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Field | mathematics | Britannica
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The first step toward answering this question was the concept of a field, defined as any subset… </p><a class="read-more" href="/science/algebra/Fundamental-concepts-of-modern-algebra#ref762852">Read More</a></li></ul></section><section><h3>algebraic geometry</h3><ul class="list-unstyled"><li><div class="md-assembly-wrapper card-media " data-type="image"><img src="https://cdn.britannica.com/54/2354-050-B47B3735/Babylonian-tablet.jpg?w=300" alt="Babylonian mathematical tablet" data-width="869" data-height="854" loading="lazy"></div> <span class="index-xref">In <a href="https://www.britannica.com/science/mathematics/Algebraic-topology#ref536622" class="md-crosslink " data-show-preview="true">mathematics: Developments in pure mathematics</a></span><p class="topic-paragraph">…of an abstract theory of fields, it was natural to want a theory of varieties defined by equations with coefficients in an arbitrary field. This was provided for the first time by the French mathematician André Weil, in his <em>Foundations of Algebraic Geometry</em> (1946), in a way that drew on…</p><a class="read-more" href="/science/mathematics/Algebraic-topology#ref536622">Read More</a></li></ul></section><section><h3>modern algebra</h3><ul class="list-unstyled"><li><div class="md-assembly-wrapper card-media " data-type="image"><img src="https://cdn.britannica.com/75/77675-004-4D35FC0A/curve.jpg" alt="algebraic equation" data-width="320" data-height="220" loading="lazy"></div> <span class="index-xref">In <a href="https://www.britannica.com/science/modern-algebra#ref790511" class="md-crosslink " data-show-preview="true">modern algebra: Fields</a></span><p class="topic-paragraph">In itself a set is not very useful, being little more than a well-defined collection of mathematical objects. However, when a set has one or more operations (such as addition and multiplication) defined for its elements, it becomes very useful. If the operations satisfy… </p><a class="read-more" href="/science/modern-algebra#ref790511">Read More</a></li></ul></section><section><h3>work of Steinitz</h3><ul class="list-unstyled"><li><div class="md-assembly-wrapper card-media " data-type="image"><img src="https://cdn.britannica.com/61/67361-050-F48C1CC9/map-millennium-mathematicians-Greco-Roman-Thales-Hypatia-Alexandria.jpg" alt="mathematicians of the Greco-Roman world" data-width="1600" data-height="887" loading="lazy"></div> <span class="index-xref">In <a href="https://www.britannica.com/science/algebra/Structural-algebra#ref762995" class="md-crosslink " data-show-preview="true">algebra: Hilbert and Steinitz</a></span><p class="topic-paragraph">…on the abstract theory of fields that was an important milestone on the road to the structural image of algebra. His work was highly structural in that he first established the simplest kinds of subfields that any field contains and established a classification system. He then investigated how properties were…</p><a class="read-more" href="/science/algebra/Structural-algebra#ref762995">Read More</a></li></ul></section></section></div> </div> </div> </div> <aside class="col-md-da-320" data-page="1"></aside> </div> </div> </div> </div> </article></div> </div></div> </div> </main> <div id="md-footer"></div> <noscript><iframe src="//www.googletagmanager.com/ns.html?id=GTM-5W6NC8" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript> <script type="text/javascript" id="_informizely_script_tag"> var IzWidget = IzWidget || {}; (function (d) { var scriptElement = d.createElement('script'); scriptElement.type = 'text/javascript'; scriptElement.async = true; scriptElement.src = "https://insitez.blob.core.windows.net/site/f780f33e-a610-4ac2-af81-3eb184037547.js"; var node = d.getElementById('_informizely_script_tag'); node.parentNode.insertBefore(scriptElement, node); } )(document); </script> <!-- Ortto ebmwprod capture code --> <script> window.ap3c = window.ap3c || {}; var ap3c = window.ap3c; ap3c.cmd = ap3c.cmd || []; ap3c.cmd.push(function() { ap3c.init('ZO4siT4cLwnykPnzZWJtd3Byb2Q', 'https://engage.email.britannica.com/'); ap3c.track({v: 0}); }); ap3c.activity = function(act) { ap3c.act = (ap3c.act || []); ap3c.act.push(act); }; var s, t; s = document.createElement('script'); s.type = 'text/javascript'; s.src = "https://engage.email.britannica.com/app.js"; t = document.getElementsByTagName('script')[0]; t.parentNode.insertBefore(s, t); </script> <script class="marketing-page-info" type="application/json"> {"pageType":"Topic","templateName":"INDEX","pageNumber":1,"pagesTotal":1,"pageId":206155,"pageLength":0,"initialLoad":true,"lastPageOfScroll":false} </script> <script class="marketing-content-info" type="application/json"> [] </script> <script src="https://cdn.britannica.com/mendel-resources/3-130/js/libs/jquery-3.5.0.min.js?v=3.130.14"></script> <script type="text/javascript" data-type="Init Mendel Code Splitting"> (function() { $.ajax({ dataType: 'script', cache: true, url: 'https://cdn.britannica.com/mendel-resources/3-130/dist/topic-page.js?v=3.130.14' }); })(); </script> <script class="analytics-metadata" type="application/json"> {"leg":"A","adLeg":"A","userType":"ANONYMOUS","pageType":"Topic","pageSubtype":null,"articleTemplateType":"INDEX","gisted":false,"pageNumber":1,"hasSummarizeButton":false,"hasAskButton":false} </script> <script type="text/javascript"> EBStat={accountId:-1,hostnameOverride:'webstats.eb.com',domain:'www.britannica.com', json:''}; </script> <script type="text/javascript"> ( function() { $.ajax( { dataType: 'script', cache: true, url: '//www.britannica.com/webstats/mendelstats.js?v=1' } ) .done( function() { try {writeStat(null,EBStat);} catch(err){} } ); })(); </script> <div id="bc-fixed-dialogue"></div> </body> </html>