 # Up and Away

In this section we survey developments over number fields inspired by the solution

of Hilbert’s Tenth Problem. We start with a review of some terms.

• • A number field K is a finite extension of Q.
• • We denote a fixed algebraic closure of Q, i.e. the field containing the roots of all polynomials with coefficients in Z, by Q.
• • A Galois closure of a number field K over Q is the smallest Galois number field containing K inside Q.
• • A totally real number field is a number field all of whose embeddings into its algebraic closure are real.
• • A ring of integers OK of a number field K is the set of all elements of the number field satisfying monic irreducible polynomials over Z or alternatively the integral closure of Z in the number field.
• • A prime of a number field K is a non-zero prime ideal of OK .If x = 0 and x e OK, then for any prime p of K there exists a non-negative integer m such that x e pm but x / pm+1. We call m the order of x at p and write m = ordp x. If y e K and y = 0, we write y = x1, where x1, x2 e OK with x1 x2 = 0, and define ordp y = ordpx1 - ordpx2. This definition is not dependent on the choice of x1 and x2 which are of course not unique. We define ordp0 = to for any prime p of K.
• • Given x e K, x = 0, for all but finitely many primes p of K we have ordpx = 0. We define a formal (finite) product and If x e OK, then d(x) = (1), the empty product. Of course the finite products of prime ideals of OK also correspond to ideals of OK. Further, finite products of prime ideals are called integral divisors and they form a semigroup under multiplication.

• • Given an element x e Q, x = 0, we write x = mm, m, n e Z, n > 0,(m, n) = 1 and define the height of x to be the max(|m|, |n|). Given z e K, where K is a number field, we consider the monic irreducible polynomial a0 + a1 T + ??? + an-1 Tn-1 + Tn of z over Q and define the height of z, denoted by h (z), to be the max(h(a;), i = 0,...,n).
• • If K isa number field of degree n over Q and a1 = id,...,an are all the embeddings of K into a fixed algebraic closure of Q and x e K, then NK/q(x) = Пn=1 oi (x).