Algebraic number field

Finite degree (and hence algebraic) field extension of the field of rational numbers

The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

Another notion needed to define algebraic number fields is vector spaces. To the extent needed here, vector spaces can be thought of as consisting of sequences (or tuples)

Working locally and using tools such as the Frobenius map, it is always possible to explicitly compute such a basis, and it is now standard for computer algebra systems to have built-in programs to do this.

We can then compute the trace and determinant with relative ease, giving the trace and norm.

The trace form derived is a bilinear form defined by means of the trace, as

The determinant of this is 1304 = 23·163, the field discriminant; in comparison the root discriminant, or discriminant of the polynomial, is 5216 = 25·163.

Altogether, there is a three-way equivalence between ultrametric absolute values, prime ideals, and localizations on a number field.

Ramification is a purely local property, i.e., depends only on the completions around the primes p and qi. The inertia group measures the difference between the local Galois groups at some place and the Galois groups of the involved finite residue fields.

Substituting x = y + 10 in the first factor g modulo 529 yields y + 191, so the valuation | y |g for y given by g is | −191 |23 = 1. On the other hand, the same substitution in h yields y2 − 161y − 161 modulo 529. Since 161 = 7 × 23,

Since possible values for the absolute value of the place defined by the factor h are not confined to integer powers of 23, but instead are integer powers of the square root of 23, the ramification index of the field extension at 23 is two.

Generally speaking, the term "local to global" refers to the idea that a global problem is first done at a local level, which tends to simplify the questions. Then, of course, the information gained in the local analysis has to be put together to get back to some global statement. For example, the notion of sheaves reifies that idea in topology and geometry.

Many results valid for function fields also hold, at least if reformulated properly, for number fields. However, the study of number fields often poses difficulties and phenomena not encountered in function fields. For example, in function fields, there is no dichotomy into non-archimedean and archimedean places. Nonetheless, function fields often serves as a source of intuition what should be expected in the number field case.