By Ivan Fesenko

Advent to algebraic quantity theory

This path (36 hours) is a comparatively straightforward direction which calls for minimum necessities from commutative algebra for its figuring out. Its first half (modules over primary perfect domain names, Noetherian modules) follows to a undeniable quantity the publication of P. Samuel "Algebraic conception of Numbers". Then integrality over earrings, algebraic extensions of fields, box isomorphisms, norms and lines are mentioned within the moment half. generally 3rd half Dedekind earrings, factorization in Dedekind jewelry, norms of beliefs, splitting of leading beliefs in box extensions, finiteness of the appropriate type staff and Dirichlet's theorem on devices are handled. The exposition occasionally makes use of equipment of presentation from the ebook of D. A. Marcus "Number Fields".

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5. Definition. Let F be of degree n over Q. Let σ1 , . . , σn be all Q -homomorphisms of F into C. Let τ: C → C be the complex conjugation. Then τ ◦ σi is a Q -homomorphism of F into C, so it is equal to certain σj . Note that σi = τ ◦ σi iff σi (F ) ⊂ R. Let r1 be the number of Q -homomorphisms of this type, say, after renumeration, σ1 , . . , σr1 . For every i > r1 we have τ ◦ σj = σj , so we can form couples (σj , τ ◦ σj ). Then n − r1 is an even number 2r2 , and r1 + 2r2 = n. Renumerate the σj ’s so that σi+r2 = τ ◦ σi for r1 + 1 i r1 + r2 .

For non-zero rational numbers a = m/n, b = m /n we get vp (ab) = vp (mm /(nn )) = vp (mm ) − vp (nn ) = vp (m) + vp (m ) − vp (n) − vp (n ) = vp (m) − vp (n) + vp (m ) − vp (n ) = vp (m/n) + vp (m /n ) = vp (a) + vp (b). Thus vp is a homomorphism from Q× to Z. 2. p -adic norm. Define the p -adic norm of a rational number α by |α|p = p−vp (α) , |0|p = 0. Then |αβ|p = |α|p |β|p . If α = m/n with integer m, n relatively prime to p, then vp (m) = vp (n) = 0 and |α|p = 1. In particular, | − 1|p = |1|p = 1 and so | − α|p = |α|p for every rational α .

Let p be an odd prime. Let F = Q(ζp ) be the p th cyclotomic field. If p doesn’t divide |CF |, or, equivalently, p does not divide numerators of (rational) Bernoulli numbers B2 , B4 , . . , Bp−3 given by t = t e −1 ∞ i=0 Bi i t, i! then the Fermat equation Xp + Y p = Zp does not have positive integer solutions. Among primes < 100 only 37, 59 and 67 don’t satisfy the condition that p does not divide |CF |, so Kummer’s theorem implies that for any other prime number smaller 100 the Fermat equation does not have positive integer solutions.