Galois theory exercises (incomplete)

Written as an undergrad introduction to Galois theory. The main focus is on intuition. I have tried to be as thorough as possible but some proofs are omitted. These notes only deal with the Galois theory of finite extensions.

This was a summer project I undertook after my 3rd undergraduate year, under the supervision of Dr. Neil Dummigan.

The book is quite nice and real understanding may could be achieved! :)

The first efficient general primality proving method was proposed in the year 1980 by Adleman, Pomerance and Rumely and it used Jacobi sums. The method was further developed by H. W. Lenstra Jr. and more of his students and the resulting primality proving algorithms are often referred to under the generic name of Cyclotomy Primality Proving (CPP). In the present paper we give an overview of the theoretical background and implementation specifics of CPP, such as we understand them in the year 2007.

Proof. Since 0 ∈ Tor(M ), Tor(M ) = ∅. Assume x, y ∈ Tor(M ). Then there are r, s ∈ R such that r = 0, s = 0, rx = 0 and sy = 0. Since R has no zero divisors, rs = 0. We have rs(x + r y) = (rs)x + (rs)r y = (sr)x + (rsr )y = s(rx) + (rr )sy = 0 for rs = 0. Therefore x + r y ∈ Tor(M ) for any x, y ∈ M, and r ∈ R.

2011

The second part of my masters dissertation, done under the supervision of Dr. Neil Dummigan. This installment proves everything done informally in the first part. This is quite a difficult and lengthy task and many new devices need to be invented, such as the ideles and the Herbrand quotient. Finally, we apply the theory to the representation of primes by the quadratic form x^2 + ny^2, giving some examples.

uni-math.gwdg.de

Abstract. Two rational primes p, q are called dual elliptic if there is an el-liptic curve E mod p with q points. They were introduced as an interesting means for combining the strengths of the elliptic curve and cyclotomy primal-ity proving algorithms. By extending to elliptic curves ...

This paper is a survey of results obtained by the present author towards proving Rubin's integral version of Stark's conjecture for abelian Lfunctions of arbitrary order of vanishing at the origin. Rubin's conjecture is stated and its links to the classical integral Stark conjecture for L-functions of order of vanishing 1 are discussed. A weaker version of Rubin's conjecture formulated by the author in [P4] is also stated and its links to Rubin's conjecture are discussed. Evidence in support of the validity of Rubin's conjecture is provided. A series of applications of Rubin's conjecture to the theory of Euler Systems, groups of special units and Gras-type conjectures are given.