Stewart and Tall present students with the fourth edition of their treatment of

algebraic number theory, FermatAEs last theorem, and a variety of related mathematical subjects.

The proofs of the theorems are completely self-contained except for several simple observations which follow from some earlier results on additive and multiplicative relations with conjugate

algebraic numbers. Specifically, we shall use the fact that, e.g., by [1, Theorem 4], for any n [greater than or equal to] 3 distinct

algebraic numbers [[alpha].sub.1], ..., [[alpha].sub.n] conjugate over Q we have

Neukirch,

Algebraic number theory, the Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 322, (1999).

If m, [[epsilon].sub.0], [beta], and [bar.[beta]] are defined as before, then A = ([beta][[epsilon].sup.m.sub.0] + [bar.[beta]][[bar.[epsilon].sub.0].sup.m])/([beta] + [bar.[beta]]) is an

algebraic number in the field Q([square root of -5]).

Our main results (Theorem 5 and Proposition 7) may seem surprising as we might expect that any

algebraic number would be computable in our setting.

Let a, b, c > 0 and u be a real

algebraic number. For x [member of] R and n [greater than or equal to] 0

Stan, Florin, University of Illinois, Urbana-Champaign, Trace problems in

algebraic number fields and applications to characters of finite groups.

Ten chapters cover

algebraic number theory and quadratic fields; ideal theory; binary quadratic forms; Diophantine approximation; arithmetic functions; p-adic analysis; Dirichlet characters, density, and primes in progression; applications to Diophantine equations; elliptic curves; and modular forms.

In these cases, [rho] is an

algebraic number of degree 2 (e.g., p = 1 + 2[square root of 2] for the step set [??]).

Even more of a challenge to come to grips with the complexities of Pythagoras and

algebraic number theory if your dad is out of work and the family diet is baked beans.

* Mathematicians reached a milestone in

algebraic number theory by proving the local Langlands correspondence, a conjecture that concerns prime numbers and perfect squares (157: 47).

An

algebraic number is one that can serve as a solution to a polynomial equation made up of x and powers of x.

He also considers fields that make use of Galois Theory, among them finite fields, permutation polynomials, and

algebraic number theory.

Neukirch,

Algebraic number theory, translated from the 1992 German original and with a note by Norbert Schappacher, Grundlehren der Mathematischen Wissenschaften, 322, Springer, Berlin, 1999.