Algebraic number field
This study reveals hidden structures behind the rational numbers, by using algebraic methods.
A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions.
Another notion needed to define algebraic number fields is vector spaces.
Furthermore, all members of any sequence can be multiplied by a single element c of the fixed field.
A field contains no zero divisors and this property is inherited by any subring, so the ring of integers of
denotes the norm of the prime ideal or, equivalently, the (finite) number of elements in the residue field
Using much more advanced techniques including algebraic K-theory and Tamagawa measures, modern number theory deals with a description, if largely conjectural (see Tamagawa number conjecture), of values of more general L-functions.
Invariants of matrices, such as the trace, determinant, and characteristic polynomial, depend solely on the field element
The integral trace form, an integer-valued symmetric matrix is defined as
that is a root of a monic polynomial with integer coefficients then the same property holds for the corresponding matrix A.
The property of being an algebraic integer is defined in a way that is independent of a choice of a basis in
This basis element induces the identity map on the 3-dimensional vector space,
Essentially, an absolute value is a notion to measure the size of elements
Two such absolute values are considered equivalent if they give rise to the same notion of smallness (or proximity).
, the following non-trivial norms occur (Ostrowski's theorem): the (usual) absolute value, sometimes denoted
, which gives rise to the complete topological field of the real numbers
Note the general situation typically considered is taking a number field
-adic numbers may similarly play the role of the rationals; in particular, we can define the norm and trace in exactly the same way, now giving functions mapping to
Yet another, equivalent way of describing ultrametric places is by means of localizations of
Altogether, there is a three-way equivalence between ultrametric absolute values, prime ideals, and localizations on a number field.
In fact, this observation is useful[6]pg 13 while looking at the base change of an algebraic field extension of
This intuition also serves to define ramification in algebraic number theory.
The connection between this definition and the geometric situation is delivered by the map of spectra of rings
Ramification is a purely local property, i.e., depends only on the completions around the primes p and qi.
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.
For more general number fields, class field theory, specifically the Artin reciprocity law gives an answer by describing Gab in terms of the idele class group.
This enables the use of group cohomology for the Galois group Gal(K), also known as Galois cohomology, which in the first place measures the failure of exactness of taking Gal(K)-invariants, but offers deeper insights (and questions) as well.
Then, of course, the information gained in the local analysis has to be put together to get back to some global statement.
A prototypical question, posed at a global level, is whether some polynomial equation has a solution in
However, the idea of passing from local data to global ones proves fruitful in class field theory, for example, where local class field theory is used to obtain global insights mentioned above.
