Algebraic Number Theory [Course Notes] by Matthew Baker

By Matthew Baker

Path Notes (Fall 2006) Math 8803, Georgia Tech, model 24 Nov 2012
Downloaded from http://people.math.gatech.edu/~mbaker/pdf/ANTBook.pdf

Show description

Read or Download Algebraic Number Theory [Course Notes] PDF

Similar nonfiction_13 books

Missionary Discourses of Difference: Negotiating Otherness in the British Empire, 1840–1900

Missionary Discourse examines missionary writings from India and southern Africa to discover colonial discourses approximately race, faith, gender and tradition. The booklet is organised round 3 issues: kinfolk, ailment and violence, which have been key parts of missionary main issue, and demanding axes round which colonial distinction was once cast.

Remote Sensing Handbook - Three Volume Set: Land Resources Monitoring, Modeling, and Mapping with Remote Sensing

A quantity within the three-volume distant Sensing guide sequence, Land assets tracking, Modeling, and Mapping with distant Sensing files the medical and methodological advances that experience taken position over the past 50 years. the opposite volumes within the sequence are Remotely Sensed facts Characterization, type, and Accuracies, and distant Sensing of Water assets, mess ups, and concrete reports.

Extra resources for Algebraic Number Theory [Course Notes]

Sample text

Let ∆ := ∆(α1 , . . , αn ), ∆ := ∆(α1 , . . , αn ). Writing each basis in terms of the other, we find that there are nonsingular n×n matrices M, M with integer coefficients such that ∆ = det(M )2 ∆ and ∆ = det(M )2 ∆. The result follows easily from this. 2, we may define the discriminant of OK (or, by abuse of terminology, the discriminant of K) to be the discriminant of any integral basis for OK . We write ∆(OK ) or ∆K for the discriminant of OK . 3. If α1 , . . , αn ∈ OK form a basis for K/Q and M denotes the Z-module spanned by α1 , .

We will sometimes write Km to denote the cyclotomic field Q(ζm ). We recall for later use that Km /Q is a Galois extension with Galois group isomorphic to (Z/mZ)∗ . The fact that Km /Q is Galois follows t from the fact that, by the irreducibility of Φm (x), ζm ∈ Km is a root of Φm (x) whenever 1 ≤ t < m and gcd(t, m) = 1. Furthermore, the map from (Z/mZ)∗ to Gal(Km /Q) sending t to the automorphism t σt : ζm → ζm is a group isomorphism. 30. Let m be a positive integer, and let d = φ(m) be the degree of the minimal polynomial Φm (x) of ζm over Q.

Proof. 15, we have OK /pi = OK /(p, gi (θ)) ∼ = Z[θ]/ (p, gi (θ)) ∼ = Z[x]/ (p, gi (x)) ∼ = Fp [x]/ gi (x) . Since gi (x) is irreducible over Fp , we know that Fp [x]/ gi (x) is a field of degree fi := deg(gi ) over Fp . Therefore pi is a prime ideal with norm pfi . 1) ei fi = deg g(x) = n. i=1 3. THE SPLITTING OF PRIMES 41 We now prove that the pi ’s are distinct. Given i = j, we know that gi (x) and gj (x) are relatively prime in Fp [x]. So there exist a(x), b(x) ∈ Z[x] such that 1 = a(x) · gi (x) + b(x) · gj (x) in Fp [x].

Download PDF sample

Rated 4.50 of 5 – based on 27 votes