1. Unique Prime Factorization of Ideals: 21 September 2010 2 2. Minkowski Theory: 28 September 2010 5 October 2010 4 3. Cyclotomic Fields and Fermat: 6 13 October 2010 9 4. Multiplicative Minkowski and Dirichlet’s Unit Theorem: 19 26 October 2010 10 5. Factoring Rational Primes in Algebraic Number Fields: 26 October 2010 14 6. Application to Zeta Functions: 3 9 November 2010 18 7. Hilbert’s Ram...

Forward We start with the set of natural numbers, N = {1, 2, 3, . . .} equipped with the familiar addition and multiplication and assume that it satisfies the induction axiom. It allows us to establish division with a residue and the Euclid’s algorithm that computes the greatest commond divisor of two natural numbers. It also leads to a proof of the fundamental theorem of arithmetic: Every natu...

We will start with introducing congruences and investigating modular arithmetic: the set Z/nZ of “integers modulo n” forms a ring. This ring is a field if and only if n is a prime number. A study of the multiplicative structure leads to Fermat’s Little Theorem (for prime n) and to the Euler phi function and Euler’s generalization of Fermat’s Theorem. Another basic tool is the Chinese Remainder ...

Historically, computation has been a driving force in the development of mathematics. To help measure the sizes of their fields, the Egyptians invented geometry. To help predict the positions of the planets, the Greeks invented trigonometry. Algebra was invented to deal with equations that arose when mathematics was used to model the world. The list goes on, and it is not just historical. If an...

s for September 6-7, 2014 Meeting

A number b divides a if the remainder is zero. We denote it by b | a. b a denotes that b does not divide a. If b divides a then a is a multiple of b. Now we can define the greatest common divisor (GCD). The GCD of two numbers a and b is defined as the biggest number which divides both a as well as b. It is also denoted by gcd(a, b). One of the important case is when gcd(a, b) = 1, i.e., there i...

Galois group, 222adeles, 149Artin’s conjecture, 293automorphic form, 292bar resolution, 172biquadratic reciprocity, 271Brauer group, 205Brauer-Hasse-Noether theorem, 252central simple algebra, 206centralizer, 206chain map, 163change of group, 184Chebotarev density theorem, 282class formation, 222class group, 33cohomological functor, 16...

What is number theory? One might have thought that it was simply the study of numbers, but that is too broad a definition, since numbers are almost ubiquitous in mathematics. To see what distinguishes number theory from the rest of mathematics, let us look at the equation x + y = 15925, and consider whether it has any solutions. One answer is that it certainly does: indeed, the solution set for...