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<!DOCTYPE html><html><head><title>The Theory of Classical Valuations - Paulo Ribenboim - Google Books</title><link rel="stylesheet" href="/books/css/_a33f2a89320471e58c940b9287b9d4eb/kl_about_this_book_kennedy_full_bundle.css" type="text/css" /><link rel="stylesheet"href="https://fonts.googleapis.com/css2?family=Product+Sans:wght@400"><script src="/books/javascript/atb_a33f2a89320471e58c940b9287b9d4eb__en.js"></script><link rel="canonical" href="https://books.google.com/books/about/The_Theory_of_Classical_Valuations.html?id=kL1HpN1KPZsC"/><meta property="og:url" content="https://books.google.com/books/about/The_Theory_of_Classical_Valuations.html?id=kL1HpN1KPZsC"/><meta name="title" content="The Theory of Classical Valuations"/><meta name="description" content="In his studies of cyclotomic fields, in view of establishing his monumental theorem about Fermat's last theorem, Kummer introduced "local" methods. They are concerned with divisibility of "ideal numbers" of cyclotomic fields by lambda = 1 - psi where psi is a primitive p-th root of 1 (p any odd prime). Henssel developed Kummer's ideas, constructed the field of p-adic numbers and proved the fundamental theorem known today. Kurschak formally introduced the concept of a valuation of a field, as being real valued functions on the set of non-zero elements of the field satisfying certain properties, like the p-adic valuations. Ostrowski, Hasse, Schmidt and others developed this theory and collectively, these topics form the primary focus of this book."/><meta property="og:title" content="The Theory of Classical Valuations"/><meta property="og:type" content="book"/><meta property="og:site_name" content="Google Books"/><meta property="og:image" content="https://books.google.com.sg/books/content?id=kL1HpN1KPZsC&printsec=frontcover&img=1&zoom=1&edge=curl&imgtk=AFLRE71xZBBo1YY59RcMfykuVZWsp3g8D3GcS5iPFaX-ENObqR78PfGjoan-PmeeKWLmoQj_nH8a3r9FsrjiJgmcoUnk33enqByOCJzW3VUvkYSx3s4JOshxHVjL7vfYnfmQHl_AraH0"/><link rel="image_src" href="https://books.google.com.sg/books/content?id=kL1HpN1KPZsC&printsec=frontcover&img=1&zoom=1&edge=curl&imgtk=AFLRE71xZBBo1YY59RcMfykuVZWsp3g8D3GcS5iPFaX-ENObqR78PfGjoan-PmeeKWLmoQj_nH8a3r9FsrjiJgmcoUnk33enqByOCJzW3VUvkYSx3s4JOshxHVjL7vfYnfmQHl_AraH0"/><script></script><style>#gbar,#guser{font-size:13px;padding-top:1px !important;}#gbar{height:22px}#guser{padding-bottom:7px !important;text-align:right}.gbh,.gbd{border-top:1px solid #c9d7f1;font-size:1px}.gbh{height:0;position:absolute;top:24px;width:100%}@media all{.gb1{height:22px;margin-right:.5em;vertical-align:top}#gbar{float:left}}a.gb1,a.gb4{text-decoration:underline !important}a.gb1,a.gb4{color:#00c !important}.gbi .gb4{color:#dd8e27 !important}.gbf .gb4{color:#900 !important} #gbar { padding:.3em .6em !important;}</style></head><body class=""><div id=gbar><nobr><a target=_blank class=gb1 href="https://www.google.com.sg/search?tab=pw">Search</a> <a target=_blank class=gb1 href="https://www.google.com.sg/imghp?hl=en&tab=pi">Images</a> <a target=_blank class=gb1 href="https://maps.google.com.sg/maps?hl=en&tab=pl">Maps</a> <a target=_blank class=gb1 href="https://play.google.com/?hl=en&tab=p8">Play</a> <a target=_blank class=gb1 href="https://www.youtube.com/?tab=p1">YouTube</a> <a target=_blank class=gb1 href="https://news.google.com/?tab=pn">News</a> <a target=_blank class=gb1 href="https://mail.google.com/mail/?tab=pm">Gmail</a> <a target=_blank class=gb1 href="https://drive.google.com/?tab=po">Drive</a> <a target=_blank class=gb1 style="text-decoration:none" href="https://www.google.com.sg/intl/en/about/products?tab=ph"><u>More</u> »</a></nobr></div><div id=guser width=100%><nobr><span id=gbn class=gbi></span><span id=gbf class=gbf></span><span id=gbe></span><a target=_top id=gb_70 href="https://www.google.com/accounts/Login?service=print&continue=https://books.google.com.sg/books%3Fid%3DkL1HpN1KPZsC%26source%3Dgbs_navlinks_s%26hl%3Den&hl=en&ec=GAZACg" class=gb4>Sign in</a></nobr></div><div class=gbh style=left:0></div><div class=gbh style=right:0></div><div role="alert" style="position: absolute; left: 0; right: 0;"><a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&source=gbs_navlinks_s&hl=en&output=html_text" title="Screen reader users: click this link for accessible mode. 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They are concerned with divisibility of "ideal numbers" of cyclotomic fields by lambda = 1 - psi where psi is a primitive <i>p</i>-th root of 1 (p any odd prime). Henssel developed Kummer's ideas, constructed the field of <i>p</i>-adic numbers and proved the fundamental theorem known today. Kurschak formally introduced the concept of a valuation of a field, as being real valued functions on the set of non-zero elements of the field satisfying certain properties, like the <i>p</i>-adic valuations. Ostrowski, Hasse, Schmidt and others developed this theory and collectively, these topics form the primary focus of this book.</div></div></div><div class="search_box_wrapper"><form action=/books id=search_form style="margin:0px;padding:0px;" method=get> <input type=hidden name="id" value="kL1HpN1KPZsC"><table cellpadding=0 cellspacing=0 class="swv-table"><tr><td class="swv-td-search"><span><input id=search_form_input type=text maxlength=1024 class="text_flat swv-input-search" aria-label="Search in this book" name=q value="" title="Search inside" accesskey=i></span></td><td class="swv-td-space"><div> </div></td><td><input type=submit value="Search inside"></td></tr></table><script type="text/javascript">if (window['_OC_autoDir']) {_OC_autoDir('search_form_input');}</script></form><div id="preview-link"><a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&printsec=frontcover" class="primary"><span dir=ltr>Preview this book</span> »</a></div></div></td> </tr></table><div id="summary-second-column"></div></div></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="selected_pages_anchor"></a>Selected pages</h3><div id=selected_pages class=about_content><div id=selected_pages_v><div class="selectedpagesthumbnail"><a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&pg=PP9&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/content?id=kL1HpN1KPZsC&pg=PP9&img=1&zoom=1&sig=ACfU3U0-0_BIyqjl0GqtmCdNgYTJKdU1RQ" alt="Title Page" title="Title Page" height="160" border="1"></a><br/><a class="primary" href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&pg=PP9&source=gbs_selected_pages&cad=1" >Title Page</a></div><div class="selectedpagesthumbnail"><a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&pg=PA395&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/content?id=kL1HpN1KPZsC&pg=PA395&img=1&zoom=1&sig=ACfU3U2047DQnqBO1TpOla2lDUF20szIKw" alt="Index" title="Index" height="160" border="1"></a><br/><a class="primary" href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&pg=PA395&source=gbs_selected_pages&cad=1" >Index</a></div><div style="clear:both;"></div></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="toc_anchor"></a>Contents</h3><div id=toc class=about_content><div id=toc_v><div class="first_toc_column"><div class="first_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Absolute Values of Fields</span></span></div></td><td class="toc_number" align=right>3</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>12 Generalities About Absolute Values of a Field</span></span></div></td><td class="toc_number" align=right>6</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>13 Absolute Values of Q</span></span></div></td><td class="toc_number" align=right>13</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>14 The Independence of Absolute Values</span></span></div></td><td class="toc_number" align=right>16</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>15 The Topology of Valued Fields</span></span></div></td><td class="toc_number" align=right>19</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>16 Archimedean Absolute Values</span></span></div></td><td class="toc_number" align=right>32</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>17 Topological Characterizations of Valued Fields</span></span></div></td><td class="toc_number" align=right>43</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Valuations of a Field</span></span></div></td><td class="toc_number" align=right>55</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><div class="second_toc_column"><div class="second_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>103 Characterizations of Valuation Domains</span></span></div></td><td class="toc_number" align=right>288</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Galois Groups of Algebraic Extensions of Infinite Degree</span></span></div></td><td class="toc_number" align=right>307</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>112 The Abelian Closure of Q</span></span></div></td><td class="toc_number" align=right>320</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Ideals Valuations and Divisors in Algebraic Extensions of Infinite Degree over Q</span></span></div></td><td class="toc_number" align=right>333</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>121 Ideals</span></span></div></td><td class="toc_number" align=right>334</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>122 Valuations Dedekind Domains and Examples</span></span></div></td><td class="toc_number" align=right>339</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>123 Divisors of Algebraic Number Fields of Infinite Degree</span></span></div></td><td class="toc_number" align=right>356</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>A Glimpse of Krull Valuations</span></span></div></td><td class="toc_number" align=right>365</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><br style="clear:both;"/></div><span onclick="_OC_setListSectionVisible('toc_h', 1)" class=morelesslink id=toc_hc0 style="display:none"><br>More</span><div id=toc_hd1><div class="first_toc_column"><div class="first_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>22 Complete Valued Fields and Qp</span></span></div></td><td class="toc_number" align=right>67</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Polynomials and Henselian Valued Fields</span></span></div></td><td class="toc_number" align=right>79</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>32 Henselian Valued Fields</span></span></div></td><td class="toc_number" align=right>94</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Extensions of Valuations</span></span></div></td><td class="toc_number" align=right>107</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>42 The Set of Extensions of a Valuation</span></span></div></td><td class="toc_number" align=right>118</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Uniqueness of Extensions of Valuations and PolyComplete Fields</span></span></div></td><td class="toc_number" align=right>127</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>52 PolyComplete Fields</span></span></div></td><td class="toc_number" align=right>146</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Extensions of Valuations Numerical Relations</span></span></div></td><td class="toc_number" align=right>151</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>62 Numerical Relations in the General Case</span></span></div></td><td class="toc_number" align=right>158</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>63 Some Interesting Examples</span></span></div></td><td class="toc_number" align=right>172</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>64 Appendix on pGroups</span></span></div></td><td class="toc_number" align=right>178</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Power Series and the Structure of Complete Valued Fields</span></span></div></td><td class="toc_number" align=right>185</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>72 Structure of Complete Discrete Valued Fields</span></span></div></td><td class="toc_number" align=right>193</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Decomposition and Inertia Theory</span></span></div></td><td class="toc_number" align=right>213</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>82 Inertia Theory</span></span></div></td><td class="toc_number" align=right>217</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Ramification Theory</span></span></div></td><td class="toc_number" align=right>227</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>92 Higher Ramification</span></span></div></td><td class="toc_number" align=right>244</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Valuation Characterizations of Dedekind Domains</span></span></div></td><td class="toc_number" align=right>263</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>102 Characterizations of Dedekind Domains</span></span></div></td><td class="toc_number" align=right>280</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><div class="second_toc_column"><div class="second_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>133 Suggestions for Further Study</span></span></div></td><td class="toc_number" align=right>372</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Commutative Fields and Characters of Finite Abelian Groups</span></span></div></td><td class="toc_number" align=right>373</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>A2 Algebraic Elements Algebraically Closed Fields</span></span></div></td><td class="toc_number" align=right>374</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>A3 Algebraic Number Fields</span></span></div></td><td class="toc_number" align=right>375</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>A5 Normal Extensions and Splitting Fields</span></span></div></td><td class="toc_number" align=right>376</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>A6 Separable Extensions</span></span></div></td><td class="toc_number" align=right>377</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>A7 Galois Extensions</span></span></div></td><td class="toc_number" align=right>378</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>A8 Roots of Unity</span></span></div></td><td class="toc_number" align=right>379</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>A9 Finite Fields</span></span></div></td><td class="toc_number" align=right>381</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>All The Discriminant</span></span></div></td><td class="toc_number" align=right>382</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>A 12 Discriminant and Resultant of Polynomials</span></span></div></td><td class="toc_number" align=right>383</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>A 13 Inseparable Extensions</span></span></div></td><td class="toc_number" align=right>385</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>A 14 Perfect Fields</span></span></div></td><td class="toc_number" align=right>386</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>A 16 Orderable Fields</span></span></div></td><td class="toc_number" align=right>387</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>A 17 The Theorem of Artin</span></span></div></td><td class="toc_number" align=right>388</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Bibliography</span></span></div></td><td class="toc_number" align=right>391</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><a class="primary" href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&pg=PA395&source=gbs_toc_r&cad=2" ><span title="Index" style="white-space:nowrap"><span dir=ltr>Index</span></span></a></div></td><td class="toc_number" align=right>395</td></tr><tr><td class="toc_border"> </td><td 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{color: #6963CC;font-size: 10.5px;}.cloud7 {color: #6057CC;font-size: 11px;}.cloud6 {color: #574BCC;font-size: 11.5px;}.cloud5 {color: #4E3DCC;font-size: 12px;}.cloud4 {color: #4632CC;font-size: 14px;}.cloud3 {color: #3D26CC;font-size: 16px;}.cloud2 {color: #341ACC;font-size: 18px;}.cloud1 {color: #2B0DCC;font-size: 20px;}.cloud0 {color: #2200CC;font-size: 22px;}.cloud {margin-top: 4px;line-height: 24px;}.cloud a {margin-right: 6px;text-decoration: none;}.cloud a:hover {text-decoration: underline;}</style><div class=cloud><a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=a%E2%82%81&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>a₁</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=algebraic+closure&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>algebraic closure</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=algebraic+extension&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>algebraic extension</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=algebraic+number+field&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>algebraic number field</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=Archimedean&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>Archimedean</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=assume&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>assume</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=Cauchy+sequence&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>Cauchy sequence</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=Chapter&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>Chapter</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=characteristic&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>characteristic</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=coefficients&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>coefficients</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=compact&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>compact</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=complete+valued+field&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>complete valued field</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=completely+integrally+closed&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>completely integrally closed</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=conjugates&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>conjugates</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=Conversely&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>Conversely</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=Dedekind+domain&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>Dedekind domain</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=deduce&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>deduce</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=defined&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>defined</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=deg(%C6%92&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>deg(ƒ</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=deg(g&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>deg(g</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=deg(h&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>deg(h</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=denote&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>denote</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=discrete+valuation&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>discrete valuation</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=distinct&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>distinct</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=divides&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>divides</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=element&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>element</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=equivalent&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>equivalent</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=extension+of+finite&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>extension of finite</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=finite+degree&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>finite degree</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=fractional+ideal&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>fractional ideal</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=Galois+extension&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>Galois extension</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=Galois+group&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>Galois group</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=Henselian&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>Henselian</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=Henselian+valued+field&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>Henselian valued field</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=homomorphism&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>homomorphism</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=hypothesis&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>hypothesis</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=inertial+degree&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>inertial degree</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=infinite&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>infinite</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=integral+domain&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>integral domain</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=integrally+closed&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>integrally closed</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=irreducible&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>irreducible</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=isomorphism&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>isomorphism</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=K%E2%82%81&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>K₁</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=Krull+valuations&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>Krull valuations</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=Lemma&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>Lemma</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=let+F&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>let F</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=Let+xe&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>Let xe</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=mapping&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>mapping</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=maximal+ideal&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>maximal ideal</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=minimal+polynomial&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>minimal polynomial</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=monic&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>monic</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=neighborhood&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>neighborhood</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=nontrivial+valuation&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>nontrivial valuation</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=nonzero+prime+ideal&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>nonzero prime ideal</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=notations&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>notations</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=p-adic+valuation&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>p-adic valuation</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=P%E2%82%81&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>P₁</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=primary+ideal&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>primary ideal</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=prime+ideal&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>prime ideal</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=Proof&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>Proof</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=prove&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>prove</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=purely+inseparable&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>purely inseparable</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=real+number&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>real number</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=relatively+prime&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>relatively prime</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=residue+field&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>residue field</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=restriction&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>restriction</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=root+of+unity&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>root of unity</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=satisfies&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>satisfies</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=separable+extension&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>separable extension</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=subfield&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>subfield</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=subring&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>subring</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=Theorem&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>Theorem</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=topology&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>topology</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=unique+extension&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>unique extension</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=valuation+ring&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>valuation ring</span></a> <a href="https://books.google.com.sg/books?id=kL1HpN1KPZsC&q=value+group&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>value group</span></a></div></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="citations_module_anchor"></a>References to this book</h3><div id=citations_module class=about_content><div id=citations_module_v><div class="left-column-outer"><div class="left-column-inner"><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=APPtnn84FMIC&source=gbs_citations_module_r&cad=5" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=APPtnn84FMIC&printsec=frontcover&img=1&zoom=5&edge=curl" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=APPtnn84FMIC&printsec=frontcover&vq=%22The+Theory+of+Classical+Valuations%22&source=gbs_citations_module_r&cad=5"><span dir=ltr>Integral Closure of Ideals, Rings, and Modules, Volume 13</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22Craig+Huneke%22" class="secondary"><span dir=ltr>Craig Huneke</span></a>,<a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22Irena+Swanson%22" class="secondary"><span dir=ltr>Irena Swanson</span></a></span><br/><span><span style="color:#99522e">Limited preview</span> - 2006</span><br/></span></div></td><td align=right></td></tr></table></div><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=zUZm5hDhAekC&source=gbs_citations_module_r&cad=5" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=zUZm5hDhAekC&printsec=frontcover&img=1&zoom=5&edge=curl" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=zUZm5hDhAekC&printsec=frontcover&vq=%22The+Theory+of+Classical+Valuations%22&source=gbs_citations_module_r&cad=5"><span dir=ltr>Hilbert's Tenth Problem: Diophantine Classes and Extensions to Global Fields</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22Alexandra+Shlapentokh%22" class="secondary"><span dir=ltr>Alexandra Shlapentokh</span></a></span><br/><span><span style="color:#99522e">Limited preview</span> - 2006</span><br/></span></div></td><td align=right></td></tr></table></div><a class="secondary" href="https://books.google.com.sg/books?q=cites:ISBN0387985255&id=kL1HpN1KPZsC&source=gbs_citations_module_r&cad=5"><span dir=ltr>All Book Search results &raquo;</span></a></div></div><div style="clear:both"></div> </div></div></div><div class=vertical_module_list_row><h3 class="about_title">Bibliographic information</h3><div class="about_content" id="metadata_content" style="padding-bottom:.3em"><div class="metadata_sectionwrap"><table id="metadata_content_table"><tr class="metadata_row"><td class="metadata_label">Title</td><td class="metadata_value"><span dir=ltr>The Theory of Classical Valuations</span><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=bibliogroup:%22Springer+Monographs+in+Mathematics%22&source=gbs_metadata_r&cad=6"><i><span dir=ltr>Springer Monographs in Mathematics</span></i></a></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Author</span></td><td class="metadata_value"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22Paulo+Ribenboim%22&source=gbs_metadata_r&cad=6"><span dir=ltr>Paulo Ribenboim</span></a></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Edition</span></td><td class="metadata_value"><span dir=ltr>illustrated</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Publisher</span></td><td class="metadata_value"><span dir=ltr>Springer Science & Business Media, 1999</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>ISBN</span></td><td class="metadata_value"><span dir=ltr>0387985255, 9780387985251</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Length</span></td><td class="metadata_value"><span dir=ltr>403 pages</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Subjects</span></td><td class="metadata_value"><div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics%22" itemprop="url" dir=ltr><span itemprop="title">Mathematics</span></a></div> › <div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+Algebra%22" itemprop="url" dir=ltr><span itemprop="title">Algebra</span></a></div> › <div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+Algebra+Abstract%22" itemprop="url" dir=ltr><span itemprop="title">Abstract</span></a></div><br><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Algebra+/+Abstract%22&source=gbs_metadata_r&cad=6"><span dir=ltr>Mathematics / Algebra / Abstract</span></a><br/><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Algebra+/+General%22&source=gbs_metadata_r&cad=6"><span dir=ltr>Mathematics / Algebra / General</span></a><br/><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Topology%22&source=gbs_metadata_r&cad=6"><span dir=ltr>Mathematics / Topology</span></a></td></tr><tr class="metadata_row"><td>&nbsp</td><td>&nbsp</td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Export Citation</span></td><td class="metadata_value"><a class="gb-button " href="https://books.google.com.sg/books/download/The_Theory_of_Classical_Valuations.bibtex?id=kL1HpN1KPZsC&output=bibtex"><span dir=ltr>BiBTeX</span></a> <a class="gb-button " href="https://books.google.com.sg/books/download/The_Theory_of_Classical_Valuations.enw?id=kL1HpN1KPZsC&output=enw"><span dir=ltr>EndNote</span></a> <a class="gb-button " href="https://books.google.com.sg/books/download/The_Theory_of_Classical_Valuations.ris?id=kL1HpN1KPZsC&output=ris"><span dir=ltr>RefMan</span></a></td></tr></table></div><div style="clear:both"></div></div></div><script>_OC_addFlags({Host:"https://books.google.com.sg/", 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