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<!doctype html public "-//w3c//dtd html 4.0 transitional//en"> <html> <head> <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"> <meta name="GENERATOR" content="Mozilla/4.75 [en] (X11; U; SunOS 5.5.1 sun4u) [Netscape]"> <meta name="Author" content="Keith Conrad"> <meta name="Description" content="expositions"> <title>Expository papers by K. Conrad</title> </head> <body text="#000000" bgcolor="#FFFFFF" link="#0000EE" vlink="#551A8B" alink="#FF0000"> <CENTER> <H1>Expository papers</H1> </CENTER> These were written up for various reasons: course handouts, notes to accompany a talk for a (mathematically) general audience, or for some other purpose that I have since forgotten. If you find typographical or other errors in these files, or have comments, please let me know. Files that are revised will be reposted without any indication that they have been changed (sorry). <table border="1" cellpadding = "8"> <tr> <td colspan="4" align="center"> Writing Proofs </td> </tr> <tr> <td align="center"> <a href="proofs/writingtips.pdf" target="_blank"> Advice on mathematical writing </a> </td> <td align="center"> <a href="proofs/induction.pdf" target="_blank"> Examples of proofs by induction </a> </td> <td align="center"> <a href="proofs/binomcoeffintegral.pdf" target="_blank"> Proofs of integrality of binomial coefficients </a> </td> <td align="center"> <a href="proofs/welldefined.pdf" target="_blank"> Well-defined functions </a> </td> </tr> <tr> <td colspan="4" align="center"> Group Theory </tr> </td> <tr> <td align="center"> <a href="grouptheory/whygroups.pdf" target="_blank"> Why groups? </a> </td> <td align="center"> <a href="grouptheory/sign.pdf" target="_blank"> Sign of permutations </a> </td> <td align="center"> <a href="grouptheory/15puzzle.pdf" target="_blank"> The Fifteen puzzle (and Rubik's cube) </a> </td> <td align="center"> <a href="grouptheory/order.pdf" target="_blank"> Order of elements </a> </td> </tr> <tr> <td align="center"> <a href="grouptheory/cyclicgp.pdf" target="_blank"> Subgroups of cyclic groups </a> </td> <td align="center"> <a href="grouptheory/countsubgpZpaZpb.pdf" target="_blank"> Subgroups of Z/(pa) × Z/(pb) </a> </td> <td align="center"> <a href="grouptheory/cyclicmodp.pdf" target="_blank"> Cyclicity of (Z/(p))× </a> </td> <td align="center"> <a href="grouptheory/coset.pdf" target="_blank"> Cosets and Lagrange's theorem </a> </td> </tr> <tr> <td align="center"> <a href="grouptheory/quotientgroups.pdf" target="_blank"> Quotient groups </a> </td> <td align="center"> <a href="grouptheory/homomorphisms.pdf" target="_blank"> Homomorphisms </a> </td> <td align="center"> <a href="grouptheory/isomorphisms.pdf" target="_blank"> Isomorphisms </a> </td> <td align="center"> <a href="grouptheory/A4noindex2.pdf" target="_blank"> No subgroup of A4 has index 2 </a> </td> </tr> <tr> <td align="center"> <a href="grouptheory/groupsorder4and6.pdf" target="_blank"> Groups of order 4 and 6 </a> </td> <td align="center"> <a href="grouptheory/group12.pdf" target="_blank"> Groups of order 12 </a> </td> <td align="center"> <a href="grouptheory/groupsp2.pdf" target="_blank"> Groups of order p2 </a> </td> <td align="center"> <a href="grouptheory/groupsp3.pdf" target="_blank"> Groups of order p3 </a> </td> </tr> <tr> <td align="center"> <a href="grouptheory/group16.pdf" target="_blank"> Groups of order 16 </a> </td> <td align="center"> <a href="grouptheory/genquat.pdf" target="_blank"> Generalized quaternions </a> </td> <td align="center"> <a href="grouptheory/genset.pdf" target="_blank"> Generating sets </a> </td> <td align="center"> <a href="grouptheory/conjclass.pdf" target="_blank"> Conjugation in a group </a> </td> </tr> <tr> <td align="center"> <a href="grouptheory/2parametergroup.pdf" target="_blank"> A 2-parameter nonabelian group </a> </td> <td align="center"> <a href="grouptheory/dihedral.pdf" target="_blank"> Dihedral groups I </a> </td> <td align="center"> <a href="grouptheory/dihedral2.pdf" target="_blank"> Dihedral groups II </a> </td> <td align="center"> Isometries of <a href="grouptheory/isometrycpx.pdf" target="_blank">the plane </a> and complex numbers </td> </tr> <tr> <td align="center"> Isometries of <a href="grouptheory/isometryR2.pdf" target="_blank">the plane </a> and linear algebra </td> <td align="center"> Isometries of <a href="grouptheory/isometryRn.pdf" target="_blank"> Rn </td> <td align="center"> <a href="grouptheory/SL(2,R).pdf" target="_blank"> SL2(R) </a> </td> <td align="center"> <a href="grouptheory/SL(2,Z).pdf" target="_blank"> SL2(Z) </a> </td> </tr> <tr> <td align="center"> <a href="grouptheory/cauchypf.pdf" target="_blank"> Proof</a> of Cauchy's theorem </td> <td align="center"> <a href="grouptheory/cauchyapp.pdf" target="_blank"> Consequences</a> of Cauchy's theorem </td> <td align="center"> <a href="grouptheory/finite-abelian.pdf" target="_blank"> Decomposition of finite abelian groups </a> </td> </td> <td align="center"> <a href="grouptheory/gpaction.pdf" target="_blank"> Group actions </a> </td> </tr> <tr> <td align="center"> <a href="grouptheory/transitive.pdf" target="_blank"> Transitive group actions </a> </td> <td align="center"> The Sylow theorems (<a href="grouptheory/sylowpf.pdf" target="_blank">proof</a>) </td> <td align="center"> <a href="grouptheory/sylowapp.pdf" target="_blank"> Consequences</a> of Sylow theorems </td> <td align="center"> <a href="grouptheory/sylowmore.pdf" target="_blank"> More </a> on the Sylow theorems </td> </tr> <tr> <td align="center"> <a href="grouptheory/allgrouporderncyclic.pdf" target="_blank"> When are all groups of order n cyclic? </a> </td> <td align="center"> <a href="grouptheory/Ansimple.pdf" target="_blank"> Simplicity of An </a> </td> <td align="center"> <a href="grouptheory/PSLnsimple.pdf" target="_blank"> Simplicity of PSLn(F) </a> </td> <td align="center"> <a href="grouptheory/charthy.pdf" target="_blank"> Characters of finite abelian groups </a> </td> </tr> <tr> <td align="center"> <a href="grouptheory/charthyshort.pdf" target="_blank"> Characters of finite abelian groups (short version) </a> </td> <td align="center"> <a href="grouptheory/semidirect-product.pdf" target="_blank"> Semidirect Products </a> </td> <td align="center"> <a href="grouptheory/subgpseries1.pdf" target="_blank"> Subgroup series I </a> </td> <td align="center"> <a href="grouptheory/subgpseries2.pdf" target="_blank"> Subgroup series II </a> </td> </tr> <tr> <td align="center"> <a href="grouptheory/splittinggp.pdf" target="_blank"> Splitting of short exact sequences for groups </a> </td> <td align="center"> <a href="grouptheory/schurzass.pdf" target="_blank"> Schur-Zassenhaus theorem </a> </td> <td align="center"> <a href="grouptheory/relativity.pdf" target="_blank"> Relativistic addition and group theory </a> </td> <td align="center"> <a href="grouptheory/CstarqZ.pdf" target="_blank"> Escher's Print Gallery </a> and quotient groups </td> </tr> <tr> <td align="center"> <a href="grouptheory/maschke.pdf" target="_blank"> Maschke's theorem over general fields </a> </td> <td align="center"> <a href="grouptheory/affineheisrep.pdf" target="_blank"> Representations of affine and Heisenberg group over finite fields </a> </td> <td align="center"> <a href="grouptheory/irrepdeg.pdf" target="_blank"> The degree may not divide the size of the group </a> </td> <td align="center"> <a href="grouptheory/wordproblem.pdf" target="_blank"> Why word problems are hard </a> </td> <td align="center"> </td> </tr> <tr> <td colspan="4" align="center"> Ring Theory </td> </tr> <tr> <td align="center"> <a href="ringtheory/divthm.pdf" target="_blank"> Division theorem in Z and R[T] </a> </td> <td align="center"> <a href="ringtheory/countroots.pdf" target="_blank"> Counting roots of polynomials </a> </td> <td align="center"> <a href="ringtheory/ringdefs.pdf" target="_blank"> Standard definitions for rings </a> </td> <td align="center"> <a href="ringtheory/ideals.pdf" target="_blank"> Notes on ideals </a> </td> </tr> <tr> <td align="center"> <a href="ringtheory/irredtestsoverQ.pdf" target="_blank"> Irreducibility tests in Q[T] </a> </td> <td align="center"> <a href="ringtheory/reducibleallp.pdf" target="_blank"> An irreducible that factors modulo all primes </a> </td> <td align="center"> <a href="ringtheory/irredselmerpoly.pdf" target="_blank"> Irreducibility of xn - x - 1 </a> </td> <td align="center"> <a href="ringtheory/gaussnormlemma.pdf" target="_blank"> The Gauss norm and Gauss's lemma </a> </td> </tr> <tr> <td align="center"> Remarks about <a href="ringtheory/euclideanrk.pdf" target="_blank"> Euclidean domains </a> </td> <td align="center"> <a href="ringtheory/noetherian-ring.pdf" target="_blank"> Noetherian rings </a> </td> <td align="center"> <a href="galoistheory/symmfunction.pdf" target="_blank"> Symmetric polynomials </a> </td> <td align="center"> <a href="ringtheory/ufdapp.pdf" target="_blank"> Applications of unique factorization </a> </td> </tr> <tr> <td align="center"> <a href="ringtheory/polynomial-properties.pdf" target="_blank"> Nilpotents, units, and zero divisors for polynomials </a> </td> <td align="center"> <a href="ringtheory/maxideal-polyring.pdf" target="_blank"> Maximal ideals in polynomial rings </a> </td> <td align="center"> <a href="ringtheory/primvector.pdf" target="_blank"> Primitive vectors and SLn </a> </td> <td align="center"> <a href="zorn1.pdf" target="_blank"> Zorn's lemma </a> (in group theory, ring theory, and linear algebra) </td> </tr> <tr> <td align="center"> <a href="ringtheory/algebras.pdf" target="_blank"> Algebras </a> </td> <td align="center"> <a href="ringtheory/quaternionalg.pdf" target="_blank"> Quaternion algebras </a> </td> <td align="center"> </td> <td align="center"> </td> </tr> <tr> <td colspan="4" align="center"> Linear/Multilinear algebra </td> </tr> <tr> <td align="center"> <a href="linmultialg/dimension.pdf" target="_blank"> Dimension </a> </td> <td align="center"> <a href="linmultialg/minpolyandappns.pdf" target="_blank"> Minimal polynomial </a> </td> <td align="center"> <a href="linmultialg/simulcomm.pdf" target="_blank"> Simultaneous commutativity of operators </a> </td> <td align="center"> <a href="linmultialg/potdiagonalizable.pdf" target="_blank"> Potentially diagonalizable operators </a> </td> </tr> <tr> <td align="center"> <a href="linmultialg/semisimple.pdf" target="_blank"> Semisimple operators </a> </td> <td align="center"> <a href="linmultialg/diffeqdim.pdf" target="_blank"> Differential equations and linear algebra </a> </td> <td align="center"> <a href="linmultialg/linearrecursion.pdf" target="_blank"> Linear recursions over all fields </a> </td> <td align="center"> <a href="linmultialg/matrixnorm.pdf" target="_blank"> The norm of a matrix </a> </td> </tr> <tr> <td align="center"> <a href="linmultialg/descentPythag.pdf" target="_blank"> Pythagorean descent  </a> </td> <td align="center"> <a href="linmultialg/pfister.pdf" target="_blank"> Pfister's theorem on sums of squares </a> </td> <td align="center"> Hurwitz's theorem on sums of squares (by <a href="linmultialg/hurwitzlinear.pdf" target="_blank"> linear algebra</a>) </td> <td align="center"> Hurwitz's theorem on sums of squares (by <a href="linmultialg/hurwitzrepnthy.pdf" target="_blank"> representation theory</a>) </td> </tr> <tr> <td align="center"> <a href="linmultialg/sumsquareQF(T).pdf" target="_blank"> Sums of squares in Q and F(T) </a> </td> <td align="center"> <a href="linmultialg/moduleintro.pdf" target="_blank"> Introduction to modules </a> </td> <td align="center"> <a href="linmultialg/modulesoverPID.pdf" target="_blank"> Modules over a PID </a> </td> <td align="center"> <a href="linmultialg/alignedbases.pdf" target="_blank"> Simultaneously aligned bases </a> </td> </tr> <tr> <td align="center"> <a href="linmultialg/stablyfree.pdf" target="_blank"> Stably free modules </a> </td> <td align="center"> <a href="linmultialg/noetherianmod.pdf" target="_blank"> Noetherian modules </a> </td> <td align="center"> <a href="linmultialg/dualmod.pdf" target="_blank"> Dual modules </a> </td> <td align="center"> <a href="linmultialg/dualspaceinfinite.pdf" target="_blank"> Infinite-dimensional dual spaces </a> </td> </tr> <tr> <td align="center"> <a href="linmultialg/bilinearform.pdf" target="_blank"> Bilinear forms </a> </td> <td align="center"> <a href="linmultialg/univid.pdf" target="_blank"> Universal identities I </a> </td> <td align="center"> <a href="linmultialg/univid2.pdf" target="_blank"> Universal identities II </a> </td> <td align="center"> <a href="linmultialg/universalmapping.pdf" target="_blank"> Universal mapping properties </a> </td> </tr> <tr> <td align="center"> <a href="linmultialg/splittingmodules.pdf" target="_blank"> Splitting of short exact sequences for modules </a> </td> <td align="center"> <a href="linmultialg/complexification.pdf" target="_blank"> Complexification </a> </td> <td align="center"> <a href="linmultialg/tensorprod.pdf" target="_blank"> Tensor products I </a> </td> <td align="center"> <a href="linmultialg/tensorprod2.pdf" target="_blank"> Tensor products II </a> </td> </tr> <tr> <td align="center"> <a href="linmultialg/extmod.pdf" target="_blank"> Exterior powers </a> </td> <td align="center"> <a href="linmultialg/extmodbaseextn.pdf" target="_blank"> Base extension and exterior powers </a> </td> <td align="center"> </td> <td align="center"> </td> </tr> <tr> <td colspan="4" align="center"> Fields and Galois theory </td> </tr> <tr> <td align="center"> <a href="galoistheory/rootirred.pdf" target="_blank"> Roots and irreducible polynomials </a> </td> <td align="center"> <a href="galoistheory/numbersoncircle.pdf" target="_blank"> Roots on a circle </a> </td> <td align="center"> <a href="galoistheory/simpleradical.pdf" target="_blank"> Simple radical extensions </a> </td> <td align="center"> <a href="galoistheory/finitefields.pdf" target="_blank"> Finite fields </a> </td> </tr> <tr> <td align="center"> <a href="galoistheory/tracenorm.pdf" target="_blank"> Trace and norm, I </a> </td> <td align="center"> <a href="galoistheory/tracenorm2.pdf" target="_blank"> Trace and norm, II </a> </td> <td align="center"> <a href="galoistheory/separable1.pdf" target="_blank"> Separable extensions </a> </td> <td align="center"> <a href="galoistheory/perfect.pdf" target="_blank"> Perfect fields </a> </td> </tr> <tr> <td align="center"> <a href="galoistheory/algclosure.pdf" target="_blank"> Constructing algebraic closures, I </a> </td> <td align="center"> <a href="galoistheory/algclosureshorter.pdf" target="_blank"> Constructing algebraic closures, II </a></td> <td align="center"> <a href="zorn2.pdf" target="_blank"> Zorn's lemma </a> (with fields) </td> <td align="center"> <a href="galoistheory/splittingfields.pdf" target="_blank"> Splitting fields </a> </td> </tr> <tr> <td align="center"> <a href="galoistheory/separable2.pdf" target="_blank"> Separable extensions and tensor products </a> </td> <td align="center"> <a href="galoistheory/splittingisom.pdf" target="_blank"> Splitting fields and tensor products </a> </td> <td align="center"> <a href="galoistheory/galoiscorr.pdf" target="_blank"> Galois correspondence </a> </td> <td align="center"> <a href="galoistheory/galoiscorrexamples.pdf" target="_blank"> Examples of Galois correspondence </a> </td> </tr> <tr> <td align="center"> <a href="galoistheory/galoisappn.pdf" target="_blank"> Applications of Galois theory </a> </td> <td align="center"> <a href="galoistheory/galoisaspermgp.pdf" target="_blank"> Galois groups as permutation groups </a> </td> <td align="center"> <a href="galoistheory/galoiscorrthms.pdf" target="_blank"> Galois correspondence theorems </a> </td> <td align="center"> <a href="galoistheory/cubicquartic.pdf" target="_blank"> Galois groups of cubics and quartics (not char. 2) </a> </td> </tr> <tr> <td align="center"> <a href="galoistheory/cubicquarticallchar.pdf" target="_blank"> Galois groups of cubics and quartics (all characteristics) </a> </td> <td align="center"> <a href="galoistheory/cyclotomic.pdf" target="_blank"> Cyclotomic extensions </a> </td> <td align="center"> <a href="galoistheory/galoisSnAn.pdf" target="_blank"> Recognizing Galois groups Sn and An </a> </td> <td align="center"> <a href="galoistheory/linearchar.pdf" target="_blank"> Linear independence of characters </a> </td> </tr> <tr> <td align="center"> <a href="galoistheory/artinschreier.pdf" target="_blank"> The Artin-Schreier theorem </a> </td> <td align="center"> <a href="galoistheory/galoisdescent.pdf" target="_blank"> Galois descent </a> </td> <td align="center"> </td> <td align="center"> </td> </tr> <tr> <td colspan="4" align="center"> Elementary number theory </td> </tr> <tr> <td align="center"> <a href="ugradnumthy/divthmZF[T].pdf" target="_blank"> The division theorem in Z and F[T] </a> </td> <td align="center"> <a href="ugradnumthy/divgcd.pdf" target="_blank"> Divisibility and greatest common divisor </a> </td> <td align="center"> <a href="ugradnumthy/divnobezout.pdf" target="_blank"> Divisibility without Bezout's identity </a> </td> <td align="center"> <a href="ugradnumthy/modarith.pdf" target="_blank"> Modular arithmetic </a> </td> </tr> <tr> <td align="center"> <a href="ugradnumthy/modarithshort.pdf" target="_blank"> Modular arithmetic (short version) </a> </td> <td align="center"> <a href="ugradnumthy/uniquefactnZF[T].pdf" target="_blank"> Unique factorization in Z and F[T] </a> </td> <td align="center"> <a href="ugradnumthy/analogypoly.pdf" target="_blank"> Analogies between Z and F[T] </a> </td> <td align="center"> <a href="ugradnumthy/universaldivtest.pdf" target="_blank"> Universal divisibility test </a> </td>  </tr> <tr> <td align="center"> <a href="ugradnumthy/pythagtriple.pdf" target="_blank"> Pythagorean triples </a> </td> <td align="center"> <a href="ugradnumthy/fermatlittletheorem.pdf" target="_blank"> Fermat's little theorem </a> </td> <td align="center"> <a href="ugradnumthy/fermattest.pdf" target="_blank"> Fermat's test </a> </td> <td align="center"> <a href="ugradnumthy/eulerthm.pdf" target="_blank"> Euler's theorem </a> </td> </tr> <tr> <td align="center"> <a href="ugradnumthy/ordersmodm.pdf" target="_blank"> Orders in modular arithmetic </a> </td> <td align="center"> <a href="ugradnumthy/RSAnotes.pdf" target="_blank"> Number theory and cryptography </a> </td> <td align="center"> <a href="ugradnumthy/crt.pdf" target="_blank"> Chinese remainder theorem </a> </td> <td align="center"> <a href="ugradnumthy/carmichaelkorselt.pdf" target="_blank"> Carmichael numbers and Korselt's criterion </a> </td> </tr> <tr> <td align="center"> <a href="ugradnumthy/minus1squaremodp.pdf" target="_blank"> When is -1 a square modulo primes? </a> </td> <td align="center"> <a href="ugradnumthy/infinitudeofprimes.pdf" target="_blank"> The infinitude of the primes </a> </td> <td align="center"> <a href="ugradnumthy/prime-patterns-1.pdf" target="_blank"> Patterns in primes </a> </td> <td align="center"> <a href="ugradnumthy/wieferich-primes.pdf" target="_blank"> Wieferich primes </a> </td> </tr> <tr> <td align="center"> <a href="ugradnumthy/squaresandinfmanyprimes.pdf" target="_blank"> Square patterns and infinitely many primes </a> </td> <td align="center"> <a href="ugradnumthy/primestopology.pdf" target="_blank"> The "topological" proof of the infinitude of primes </a> </td> <td align="center"> <a href="ugradnumthy/fermat-numbers-and-factors.pdf" target="_blank"> Fermat numbers and their factors </a> </td> <td align="center"> <a href="ugradnumthy/perfect-numbers-mersenne.pdf" target="_blank"> Perfect numbers and Mersenne primes </a> </td> </tr> <tr> <td align="center"> <a href="ugradnumthy/lucas-lehmer-test.pdf" target="_blank"> The Lucas-Lehmer test </a> </td> <td align="center"> <a href="ugradnumthy/solovaystrassen.pdf" target="_blank"> Solovay-Strassen test </a> </td> <td align="center"> <a href="ugradnumthy/millerrabin.pdf" target="_blank"> Miller-Rabin test </a> </td> <td align="center"> <a href="ugradnumthy/irredtestFpT.pdf" target="_blank"> Irreducibility tests in Fp[T] </a> </td> </tr> <tr> <td align="center"> <a href="ugradnumthy/pelleqn1.pdf" target="_blank"> Pell's equation, I </a> </td> <td align="center"> <a href="ugradnumthy/pelleqn2.pdf" target="_blank"> Pell's equation, II </a> </td> <td align="center"> <a href="ugradnumthy/contfrac-neg-invert.pdf" target="_blank"> Negation and inversion of continued fractions </a> </td> <td align="center"> <a href="ugradnumthy/Zinotes.pdf" target="_blank"> Gaussian integers </a> </td> </tr> <tr> <td align="center"> <a href="ugradnumthy/quadraticundergrad.pdf" target="_blank"> Factoring in quadratic fields </a> </td> <td align="center"> <a href="ugradnumthy/Picksumofsq.pdf" target="_blank"> Sums of two squares and lattices </a> </td> <td align="center"> <a href="ugradnumthy/descent.pdf" target="_blank"> Proofs by descent </a> </td> <td align="center"> An example of <a href="ugradnumthy/descentbyeuler.pdf" target="_blank"> descent</a> by Euler </td> </tr> <tr> <td align="center"> <a href="ugradnumthy/congnumber.pdf" target="_blank"> Congruent number problem </a> </td> <td align="center"> <a href="ugradnumthy/3squarearithprog.pdf" target="_blank"> Arithmetic progressions of three squares </a> </td> <td align="center"> <a href="ugradnumthy/4squarearithprog.pdf" target="_blank"> Arithmetic progressions of four squares </a> </td> <td align="center"> <a href="ugradnumthy/QuadraticResiduePatterns.pdf" target="_blank"> Quadratic residue patterns modulo a prime </a> </td> </tr> <tr> <td align="center"> Quadratic reciprocity in <a href="ugradnumthy/QRcharp.pdf" target="_blank"> odd characteristic</a> </td> <td align="center"> Quadratic reciprocity in <a href="ugradnumthy/QRchar2.pdf" target="_blank"> characteristic 2 </a> </td> <td align="center"> </td> <td align="center"> </td> </tr> <tr> <td colspan="4" align="center"> Algebraic number theory </td> </tr> <tr> <td align="center"> Examples of <a href="gradnumthy/mordelleqn1.pdf" target="_blank"> Mordell's equation </a> </td> <td align="center"> <a href="gradnumthy/quadraticgrad.pdf" target="_blank"> Factoring in quadratic fields </a> </td> <td align="center"> <a href="gradnumthy/idealfactor.pdf" target="_blank"> Unique factorization of ideals </a> </td> <td align="center"> <a href="gradnumthy/dedekindf.pdf" target="_blank"> Factoring ideals after Dedekind </a> </td> </tr> <tr> <td align="center"> <a href="gradnumthy/dedekind-index-thm.pdf" target="_blank"> Dedekind's index theorem </a> </td> <td align="center"> <a href="gradnumthy/disc.pdf" target="_blank"> Discriminants and ramified primes </a> </td> <td align="center"> <a href="gradnumthy/totram.pdf" target="_blank"> Totally ramified primes and Eisenstein polynomials </a> </td> <td align="center"> <a href="gradnumthy/nopowerbasis.pdf" target="_blank"> Rings of integers without a power basis </a> </tr> <tr> </td><td align="center"> <a href="gradnumthy/integersradical.pdf" target="_blank"> The ring of integers in a radical extension </a> </td> <td align="center"> <a href="gradnumthy/notfree.pdf" target="_blank"> A non-free relative integral extension </a> </td> <td align="center"> <a href="gradnumthy/classgroupKronecker.pdf" target="_blank"> Ideal classes and Kronecker bound </a> </td> <td align="center"> <a href="gradnumthy/classgpex.pdf" target="_blank"> Class group calculations by Minkowski bound </a> </td> </tr> <tr> <td align="center"> <a href="gradnumthy/relativeintandidealclasses.pdf" target="_blank"> Ideal classes and relative integers </a> </td> <td align="center"> <a href="gradnumthy/SL2classno.pdf" target="_blank"> Ideal classes and SL2 </a> </td> <td align="center"> <a href="gradnumthy/matrixconj.pdf" target="_blank"> Ideal classes and matrix conjugation over Z </a> </td> <td align="center"> <a href="gradnumthy/unittheorem.pdf" target="_blank"> Dirichlet's unit theorem </a> </td> </tr> <tr> <td align="center"> <a href="gradnumthy/chevalleyunit.pdf" target="_blank"> Chevalley's unit theorem </a> </td> <td align="center"> <a href="gradnumthy/frobeniuspf.pdf" target="_blank"> Existence of Frobenius elements (d'après Frobenius) </a> </td> </td> <td align="center"> <a href="gradnumthy/galois-Q-factor-mod-p.pdf" target="_blank"> Galois groups over Q and factorizations mod p </a> </td> <td align="center"> <a href="gradnumthy/dedekind-galois-res-field.pdf" target="_blank"> Dedekind domains and Galois residue field extensions </a> </td> </tr> <tr> </td> <td align="center"> <a href="gradnumthy/chebappn.pdf" target="_blank"> Primes of degree 1 and congruence conditions </a> </td> <td align="center"> <a href="gradnumthy/dirichleteuclid.pdf" target="_blank"> Euclidean proofs of Dirichlet's theorem </a> </td> <td align="center"> <a href="gradnumthy/schurtheorem.pdf" target="_blank"> Irreducibility of truncated exponentials </a> </td> <td align="center"> <a href="gradnumthy/galoisselmerpoly.pdf" target="_blank"> The Galois group of xn - x - 1 over Q </a> </td> </tr> <tr> <td align="center"> <a href="gradnumthy/different.pdf" target="_blank"> The different ideal </a> </td> <td align="center"> <a href="gradnumthy/conductor.pdf" target="_blank"> The conductor ideal of an order </a> </td> <td align="center"> <a href="gradnumthy/Gauss-Jacobi-sums.pdf" target="_blank"> Gauss and Jacobi sums on finite fields and Z/mZ </a> </td> <td align="center"> <a href="gradnumthy/LfunctionGaussJacobi.pdf" target="_blank"> L-functions for Gauss and Jacobi sums </a> </td> </tr> <tr> <td align="center"> <a href="gradnumthy/Qw2.pdf" target="_blank"> Invariants of the splitting field of a cubic, I </a> </td> <td align="center"> <a href="gradnumthy/Qw3.pdf" target="_blank"> Invariants of the splitting field of a cubic, II </a> </td> <td align="center"> <a href="gradnumthy/Qw5.pdf" target="_blank"> Invariants of the splitting field of a cubic, III </a> </td> <td align="center"> <a href="gradnumthy/Qw6.pdf" target="_blank"> Invariants of the splitting field of a cubic, IV </a> </td> </tr> <tr> <td align="center"> <a href="gradnumthy/Qw7.pdf" target="_blank"> Invariants of the splitting field of a cubic, V </a> </td> <td align="center"> Ostrowski's theorem for <a href="gradnumthy/ostrowskiQ.pdf" target="_blank"> Q </a> </td> <td align="center"> Ostrowski's theorem for <a href="gradnumthy/ostrowskiQ(i).pdf" target="_blank"> Q(i) </a> </td> <td align="center"> Ostrowski's theorem for <a href="gradnumthy/ostrowskiF(T).pdf" target="_blank"> F(T) </a> </td> </tr> <tr> <td align="center"> Ostrowski's theorem for <a href="gradnumthy/ostrowskinumbfield.pdf" target="_blank"> number fields </a> </td> <td align="center"> <a href="gradnumthy/rationalsinQp.pdf" target="_blank"> The p-adic expansion of rational numbers </a> </td> <td align="center"> <a href="gradnumthy/binomialcoeffpadic.pdf" target="_blank"> Binomial coefficients and p-adic limits </a> </td> <td align="center"> <a href="gradnumthy/padicharmonicsum.pdf" target="_blank"> p-adic harmonic sums </a> </td> </tr> <tr> <td align="center"> <a href="gradnumthy/hensel.pdf" target="_blank"> Hensel's lemma </a> </td> <td align="center"> <a href="gradnumthy/multivarhensel.pdf" target="_blank"> A multivariable Hensel's lemma </a> </td> <td align="center"> <a href="gradnumthy/equivabsvalues.pdf" target="_blank"> Equivalence of absolute values </a> </td> <td align="center"> <a href="gradnumthy/equivnorms.pdf" target="_blank"> Equivalence of norms </a> </td> </tr> <tr> <td align="center"> <a href="gradnumthy/localglobal.pdf" target="_blank"> The local-global principle </a> </td> <td align="center"> <a href="gradnumthy/primepowerunitsandGLnQ.pdf" target="_blank"> Prime-power units and finite subgroups of GLn(Q) </a> </td> <td align="center"> <a href="gradnumthy/characterQ.pdf" target="_blank"> The character group of Q </a> </td> <td align="center"> <a href="gradnumthy/autRandQp.pdf" target="_blank"> Field automorphisms of R and Qp </a> </td> </tr> <tr> <td align="center"> <a href="gradnumthy/infseriespadic.pdf" target="_blank"> Infinite series in p-adic fields </a> </td> <td align="center"> <a href="gradnumthy/mahlerexpansions.pdf" target="_blank"> Mahler expansions </a> </td> <td align="center"> <a href="gradnumthy/strassmannapplication.pdf" target="_blank"> An application of Strassmann's theorem </a> </td> <td align="center"> <a href="gradnumthy/x3-2y3=1.pdf" target="_blank"> Integral solutions of x3 - 2y3 = 1. </a> </td> </tr> <tr> <td align="center"> <a href="gradnumthy/AHrootofunity.pdf" target="_blank"> Truncated Artin-Hasse series and roots of unity </a> </td> <td align="center"> <a href="http://kconrad.math.uconn.edu/blurbs/gradnumthy/maxcompact.pdf" target="_blank"> Maximal compact subgroups of GLn(Qp) </a> </td> <td align="center"> <a href="gradnumthy/GLnQpbar.pdf" target="_blank"> Compact subgroups of GLn(Qp) </a> </td> <td align="center"> <a href="http://kconrad.math.uconn.edu/blurbs/gradnumthy/sepfield-and-insep-resfield.pdf" target="_blank"> A separable extension with inseparable residue field </a> </td> </tr> <tr> <td align="center"> <a href="gradnumthy/loccptascoli.pdf" target="_blank"> Local compactness of Pontryagin dual group </a> </td> <td align="center"> <a href="gradnumthy/selmerexample.pdf" target="_blank"> Selmer's example </a> </td> <td align="center"> <a href="gradnumthy/kummer.pdf" target="_blank"> Kummer's lemma </a> </td> <td align="center"> <a href="gradnumthy/fltreg.pdf" target="_blank"> Fermat's last theorem for regular primes </a> </td> </tr> <tr> <td align="center"> <a href="gradnumthy/carlitz.pdf" target="_blank"> Carlitz extensions </a> </td> <td align="center"> <a href="gradnumthy/cfthistory.pdf" target="_blank"> History of class field theory </a> </td> <td align="center"> </td> <td align="center"> </td> </tr> <tr> <td colspan="4" align="center"> Analysis </td> </tr> <tr> <td align="center"> <a href="analysis/MVT-integral-powers.pdf" target="_blank"> The Mean Value Theorem and integral powers </a> </td> <td align="center"> <a href="analysis/growth.pdf" target="_blank"> Orders of growth </a> </td> <td align="center"> <a href="analysis/sumest.pdf" target="_blank"> Estimating growth of divergent series </a> </td> <td align="center"> <a href="analysis/asymp.pdf" target="_blank"> Asymptotic growth </a> </td> </tr> <tr> <td align="center"> <a href="analysis/stirling.pdf" target="_blank"> Stirling's formula </a> </td> <td align="center"> <a href="analysis/series.pdf" target="_blank"> Infinite series </a> </td> <td align="center"> <a href="analysis/gaussianintegral.pdf" target="_blank"> The Gaussian integral </a> </td> <td align="center"> <a href="analysis/integral-e-t2.pdf" target="_blank"> Estimating definite integrals of e-t2 in two ways </a> </td> </tr> <tr> <td align="center"> <a href="analysis/logarctan.pdf" target="_blank"> The logarithm and arctangent </a> </td> <td align="center"> <a href="analysis/TaylorRemainder.pdf" target="_blank"> The remainder in Taylor series </a> </td> <td align="center"> <a href="analysis/abelthm.pdf" target="_blank"> Abel's theorem </a> </td> <td align="center"> <a href="analysis/series_acceleration.pdf" target="_blank"> Accelerating convergence of series </a> </td> </tr> <tr> <td align="center"> <a href="analysis/arclengthIntbypartsPi.pdf" target="_blank"> Arc length, integration by parts, and π </a> </td> <td align="center"> <a href="analysis/irrational.pdf" target="_blank"> Irrationality of π and e </a> </td> <td align="center"> <a href="analysis/transcendence-e.pdf" target="_blank"> Transcendence of e </a> </td> <td align="center"> <a href="analysis/diffunderint.pdf" target="_blank"> Differentiation under the integral sign </a> </td> </tr> <tr> <td align="center"> <a href="analysis/entropypost.pdf" target="_blank"> Probability distributions and maximum entropy </a> </td> <td align="center"> <a href="analysis/metricspaces.pdf" target="_blank"> Metric spaces </a> </td> <td align="center"> <a href="analysis/contraction.pdf" target="_blank"> The contraction mapping theorem </a> </td> <td align="center"> <a href="analysis/contraction2.pdf" target="_blank"> The contraction mapping theorem, II </a> </td> </tr> <tr> <td align="center"> <a href="analysis/sequencespaceto0.pdf" target="_blank"> The space c0(K) </a> </td> <td align="center"> <a href="analysis/lpspace.pdf" target="_blank"> Lp spaces for 0 < p < 1 </a> </td> <td align="center"> </td> <td align="center"> </td> </tr> <tr> <td colspan="4" align="center"> Fundamental Theorem of Algebra </td> </tr> <tr> <td align="center"> <a href="fundthmalg/fundthmalgcalculus.pdf" target="_blank"> Proof by multivariable calculus </a> </td> <td align="center"> <a href="fundthmalg/fundthmalglinear.pdf" target="_blank"> Proof by linear algebra </a> </td> <td align="center"> <a href="fundthmalg/propermaps.pdf" target="_blank"> Proof by proper maps </a> </td> <td align="center"> </td> </tr> <tr> <td colspan="4" align="center"> Topology </td> </tr> <tr> <td align="center"> <a href="topology/connnotpathconn.pdf" target="_blank"> Spaces that are connected but not path connected </a> </td> <td align="center"> <a href="topology/finite-dim-TVS.pdf" target="_blank"> Finite-dimensional topological vector spaces </a> </td> <td align="center"> </td> <td align="center"> </td> </tr> </table>  <script type="text/javascript" language="javascript"> var sc_project=790751; var sc_partition=6; var sc_security="2ae0f47b"; var sc_invisible=1; </script> <script type="text/javascript" language="javascript" src="http://www.statcounter.com/counter/counter.js"></script><noscript><a 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