Algebraic Number Theory - James Milne

1 AlgebraicNumber MilneVersion 18, 2017An Algebraic Number field is a finite extension ofQ; an Algebraic Number is an elementof an Algebraic Number field. Algebraic Number Theory studies the arithmetic of algebraicnumber fields the ring of integers in the Number field, the ideals and units in the ring ofintegers, the extent to which unique factorization holds, and so abelian extension of a field is a Galois extension of the field with abelian Galoisgroup. Class field Theory describes the abelian extensions of a Number field in terms of thearithmetic of the notes are concerned with Algebraic Number Theory , and the sequel with class information@misc{milneANT,author={ Milne , James S.},title={ Algebraic Number Theory ( )},year={2017},note={Available at },pages={165}} (August 14, 1996). First version on the (August 31, 1998). Fixed many minor errors; added exercises and an index; 138 (February 11, 2008). Corrected; revisions and additions; 163 (September 28, 2008). Fixed problem with hyperlinks; 163 (April 30, 2009).

2 Fixed many minor errors; changed chapter and page styles; (May 29, 2011). Minor fixes; 167 (April 12, 2012). Minor (March 21, 2013). Minor (May 28, 2014). Minor fixes; 164 (March 18, 2017). Minor fixes; 165 at send comments and corrections to me at photograph is of the Fork Hut, Huxley Valley, New 1996 2017 paper copies for noncommercial personal use may be made without explicit permissionfrom the copyright ..4 Introduction ..7 Exercises ..121 Preliminaries from Commutative Algebra14 Basic definitions ..14 Ideals in products of rings ..15 Noetherian rings ..15 Noetherian modules ..16 Local rings ..17 Rings of fractions ..18 The Chinese remainder theorem ..19 Review of tensor products ..21 Exercise ..242 Rings of Integers25 First proof that the integral elements form a ring ..25 Dedekind s proof that the integral elements form a ring ..26 Integral elements ..28 Review of bases ofA-modules ..31 Review of norms and traces ..31 Review of bilinear forms.

3 32 Discriminants ..33 Rings of integers are finitely generated ..35 Finding the ring of integers ..37 Algorithms for finding the ring of integers ..40 Exercises ..443 Dedekind Domains; Factorization45 Discrete valuation rings ..45 Dedekind domains ..47 Unique factorization of ideals ..48 The ideal class group ..51 Discrete valuations ..54 Integral closures of Dedekind domains ..55 Modules over Dedekind domains (sketch)..56 Factorization in extensions ..57 The primes that ramify ..591 Finding factorizations ..61 Examples of factorizations ..62 Eisenstein extensions ..64 Exercises ..664 The Finiteness of the Class Number67 Norms of ideals ..67 Statement of the main theorem and its consequences ..69 Lattices ..72 Some calculus ..76 Finiteness of the class Number ..79 Binary quadratic forms ..80 Exercises ..825 The Unit Theorem84 Statement of the theorem ..84 Proof thatUKis finitely generated ..86 Computation of the rank ..87S-units.

4 89 Example: CM fields ..89 Example: real quadratic fields ..90 Example: cubic fields with negative discriminant ..91 Finding .K/..92 Finding a system of fundamental units ..92 Regulators ..93 Exercises ..936 Cyclotomic Extensions; Fermat s Last basic results ..94 Class numbers of cyclotomic fields .. 100 Units in cyclotomic fields .. 100 The first case of Fermat s last theorem for regular primes .. 101 Exercises .. 1037 Absolute Values; Local Fields104 Absolute Values .. 104 Nonarchimedean absolute values .. 105 Equivalent absolute values .. 106 Properties of discrete valuations .. 108 Complete list of absolute values for the rational numbers .. 108 The primes of a Number field .. 110 The weak approximation theorem .. 112 Completions .. 113 Completions in the nonarchimedean case .. 114 Newton s lemma .. 118 Extensions of nonarchimedean absolute values .. 122 Newton s polygon .. 123 Locally compact fields .. 125 Unramified extensions of a local field.

5 126 Totally ramified extensions ofK.. 128 Ramification groups .. 129 Krasner s lemma and applications .. 130 Exercises .. 1328 Global Fields134 Extending absolute values .. 134 The product formula .. 136 Decomposition groups .. 138 The Frobenius element .. 140 Examples .. 142 Computing Galois groups (the hard way) .. 143 Computing Galois groups (the easy way) .. 143 Applications of the Chebotarev density theorem .. 148 Finiteness Theorems .. 150 Exercises .. 151 ASolutions to the Exercises152 BTwo-hour use the standard (Bourbaki) notations:NDf0;1;2;:::g;ZDring of integers;RDfieldof real numbers;CDfield of complex numbers;FpDZ=pZDfield withpelements,paprime integersmandn,mjnmeans thatmdividesn, ,n2mZ. Throughout the notes,pis a prime Number , ,pD2;3;5;:::.Given an equivalence relation, denotes the equivalence class containing . The emptyset is denoted by;. The cardinality of a setSis denoted byjSj(sojSjis the Number ofelements inSwhenSis finite).

6 LetIandAbe sets; a family of elements ofAindexed byI, , is a functioni7!aiWI! Y Xis a subset ofY(not necessarily proper);XdefDY Xis defined to beY, or equalsYby definition;X Y Xis isomorphic toY;X'Y XandYare canonically isomorphic (or there is a given or unique isomorphism);,!denotes an injective map; denotes a surjective question nnnnn in is standard to use Gothic (fraktur) letters for ideals:a b c m n p q A B C M N P Qa b c m n p q A B C M N P QPrerequisitesThe algebra usually covered in a first-year graduate course, for example, Galois Theory , grouptheory, and multilinear algebra. An undergraduate Number Theory course will also be addition to the references listed at the end and in footnotes, I shall refer to the followingof my course notes (available at ):FTFields and Galois Theory , , Theory , , Field Theory , , thank the following for providing corrections and comments for earlier versions of thesenotes: Vincenzo Acciaro; Michael Adler; Giedrius Alkauskas; Baraksha; Francesc Castell`a;Kwangho Choiy; Dustin Clausen; Keith Conrad; Edgar Costa, Paul Federbush; Georg Hein;Florian Herzig; Dieter Hink; Hau-wen Huang; Enis Kaya; Keenan Kidwell; Roger Lipsett;Loy Jiabao, Jasper; Lee M.

7 Goswick; Samir Hasan; Lawrence Howe; Lars Kindler; FranzLemmermeyer; Siddharth Mathur; Bijan Mohebi; Yogesh More; Scott Mullane; SafakOzden; Wai Yan Pong; Nicol as Sirolli; Sam Spiro; Thomas Stoll; Bhupendra Nath Tiwari;Vishne Uzi; and an open source computer algebra system freely available from FERMAT(1601 1665). Stated his last theorem , and proved it formD4. He also posedthe problem of finding integer solutions to the equation,X2 AY2D1; A2Z;(1)which is essentially the problem1of finding the units inZ pA . The English mathematiciansfound an algorithm for solving the problem, but neglected to prove that the algorithm (1707 1783). He introduced analysis into the study of the prime numbers, and hediscovered an early version of the quadratic reciprocity (1736 1813). He found the complete form of the quadratic reciprocity law: pq qp D. 1/.p 1/.q 1/=4; p;qodd primes,and he proved that the algorithm for solving (1) always leads to a solution,LEGENDRE(1752 1833). He introduced the Legendre symbol mp , and gave an incom-plete proof of the quadratic reciprocity law.

8 He proved the following local-global principlefor quadratic forms in three variables overQ: a quadratic ;Y;Z/has a nontrivialzero inQif and only if it has one inRand the congruenceQ 0modpnhas a nontrivialsolution for (1777 1855). He found the first complete proofs of the quadratic reciprocity studied the Gaussian integersZ i in order to find a quartic reciprocity law. He studiedthe classification of binary quadratic forms overZ, which is closely related to the problemof finding the class numbers of quadratic (1805 1859). He introducedL-series, and used them to prove an analytic for-mula for the class Number and a density theorem for the primes in an arithmetic proved the following unit theorem : let be a root of a monic irreducible integer coefficients; suppose roots and2scomplex roots;thenZ is a finitely generated group of rankrCs (1810 1893). He made a deep study of the arithmetic of cyclotomic fields,motivated by a search for higher reciprocity laws, and showed that unique factorization couldbe recovered by the introduction of ideal numbers.

9 He proved that Fermat s last theoremholds for regular (1822 1901). He made important contributions to quadratic forms, and he showedthat the roots of a polynomial of degree5can be expressed in terms of elliptic (1823 1852). He published the first complete proofs for the cubic and quarticreciprocity (1823 1891). He developed an alternative to Dedekind s ideals. He also hadone of the most beautiful ideas in mathematics for generating abelian extensions of numberfields (the Kronecker liebster Jugendtraum).RIEMANN(1826 1866). Studied the Riemann zeta function, and made the Riemann Indian mathematician Bhaskara (12th century) knew general rules for finding solutions to the (1831 1916). He laid the modern foundations of Algebraic Number Theory byfinding the correct definition of the ring of integers in a Number field, by proving that idealsfactor uniquely into products of prime ideals in such rings, and by showing that, moduloprincipal ideals, they fall into finitely many classes.

10 Defined the zeta function of a (1842 1913). Made important progress in class field Theory and the (1861 1941). He gave the first definition of the field ofp-adic numbers (as the setof infinite sumsP1nD kanpn,an2f0;1;:::;p 1g).HILBERT(1862 1943). He wrote a very influential book on Algebraic Number Theory in1897, which gave the first systematic account of the Theory . Some of his famous problemswere on Number Theory , and have also been (1875 1960). He proved the fundamental theorems of abelian class field Theory , asconjectured by Weber and (1882 1935). Together with Artin, she laid the foundations of modern algebrain which axioms and conceptual arguments are emphasized, and she contributed to theclassification of central simple algebras over Number (1887 1947). Introduced HeckeL-series generalizing both Dirichlet sL-series andDedekind s zeta (1898 1962). He found the Artin reciprocity law , which is the main theorem ofclass field Theory (improvement of Takagi s results).