By Ivan Fesenko

Advent to algebraic quantity theory

This path (36 hours) is a comparatively easy direction which calls for minimum must haves from commutative algebra for its realizing. Its first half (modules over central excellent domain names, Noetherian modules) follows to a undeniable volume the publication of P. Samuel "Algebraic concept of Numbers". Then integrality over jewelry, algebraic extensions of fields, box isomorphisms, norms and lines are mentioned within the moment half. in general 3rd half Dedekind jewelry, factorization in Dedekind earrings, norms of beliefs, splitting of major beliefs in box extensions, finiteness of the right type workforce 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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E. k is the power of p in the factorization of m. Then vp (m1 m2 ) = vp (m1 ) + vp (m2 ). Extend vp to rational numbers putting vp (0) := ∞ and vp (m/n) = vp (m) − vp (n), this does not depend on the choice of a fractional representation: if m/n = m /n then mn = m n, hence vp (m) + vp (n ) = vp (m ) + vp (n) and vp (m) − vp (n) = vp (m ) − vp (n ). Thus we get the p -adic valuation vp : Q → Z ∪ {+∞}. 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).

Since for u ∈ U the norm NF/Q (u) = σi (u), as the product of units, is a unit in Z, it is equal to ±1. Then |σi (u)| = 1 and log |σ1 (u)| + · · · + log |σr1 (u)| + 2 log(|σr1 +1 (u)| ) + · · · + log(|σr1 +r2 (u)|2 ) = 0. We deduce that the image g(U ) is contained in the hyperplane H ⊂ Rr1 +r2 defined by the equation y1 + · · · + yr1 +r2 = 0. Since g −1 (Z) is finite for every bounded set Z , the intersection g(U ) ∩ Z is finite. 2 g(U ) has a Z -basis {yi } consisting of m r1 + r2 − 1 linearly independent vectors over Z.

So we 32 Alg number theory need to look at prime integer numbers not greater than 7 and their prime ideal divisors as potential candidates for non-principal ideals. 9 it remains prime in OQ(√−19) . 9 they split in OQ(√−19) . It is easy to check that √ √ 5 = (1 + −19)/2 (1 − −19)/2 , √ √ 7 = (3 + −19)/2 (3 − −19)/2 .. 5, since its norm is a prime number. So prime ideal factors of (5), (7) are principal ideals. 9. Thus, OQ(√−19) is a principal ideal domain. Remark. The bound given by c is not good in practical applications.