Algebraic number theory (Math 784) by Filaseta M.

By Filaseta M.

Show description

Read Online or Download Algebraic number theory (Math 784) PDF

Best algebra books

Schaum's Outlines of Linear Algebra (5th Edition)

Tricky attempt Questions? neglected Lectures? now not sufficient Time? thankfully, there's Schaum's. This all-in-one-package comprises 612 absolutely solved difficulties, examples, and perform routines to sharpen your problem-solving abilities. Plus, you could have entry to twenty-five particular video clips that includes Math teachers who clarify the best way to clear up the main quite often verified problems--it's similar to having your personal digital instruct!

Extra resources for Algebraic number theory (Math 784)

Example text

Assume a2 ≡ −1 (mod p) holds for some integer a. Since (p − 1)/2 is odd, ap−1 ≡ (a2 )(p−1)/2 ≡ (−1)(p−1)/2 ≡ −1 (mod p). Fermat’s Little Theorem implies ap−1 is either 0 or 1 modulo p, giving a contradiction. Hence, the lemma follows. Proof of Theorem 62. Assume there are integers x and y satisfying y 2 + 5 = x3 . Then y 2 ≡ x3 − 1 (mod 4) implies that x ≡ 1 (mod 4) (anything else leads to a contradiction). Observe that y 2 + 4 = x3 − 1 = (x − 1)(x2 + x + 1) 49 and x2 + x + 1 ≡ 3 (mod 4). It follows that there must be a prime p ≡ 3 (mod 4) dividing x2 + x + 1 and, hence, y 2 + 4.

Det   b1  .  .. 0 0 1 .. ... .. 0 0 .. ... .. b2 .. ... . θ .. ... . 0 0  0 0  ..  .   = θ, bn   ..  .  1 the lemma implies that ∆(ω (1) , . . , ω (n) ) = θ2 ∆(ω (1) , . . , ω (n) ). Thus, 0 < |∆(ω (1) , . . , ω (n) )| < |∆(ω (1) , . . , ω (n) )|. On the other hand, since ω (k) = β − uω (k) , each ω (j) is an algebraic integer for 1 ≤ j ≤ n. This contradicts the minimality of |∆(ω (1) , . . , ω (n) )|, completing the proof. Homework: (1) Let ω (1) , . . , ω (n) be an integral basis in Q(α).

A > 0 and b > 0. Since a + b N < x + y Now, a + b N > 1 implies 1 1 N , the minimality √ condition on x1 + y1 N now gives a contradiction. The theorem follows. √ Comment: If N is squarefree and N ≡ 1 (mod 4), then the number x + y N is called 1 1 √ the fundamental unit for the ring of algebraic integers in Q( N ). Theorem 33 implies that the fundamental generates all units in the ring in the sense that the units are given √ unit m by ±(x1 + y1 N ) where m denotes an arbitrary integer. • An example.

Download PDF sample

Rated 4.60 of 5 – based on 41 votes
This entry was posted in Algebra.