Field trace
Encyclopedia
In mathematics
, the field trace is a function
defined with respect to a finite field extension
L/K. It is a K-linear map from L to K. As an example, if L/K is a Galois extension
and α is in L, then the trace of α is the sum of all the Galois conjugates of α, i.e.
where Gal(L/K) denotes the Galois group
of L/K.
For a general finite extension L/K, the trace of an element α can be defined as the trace
of the K-linear map "multiplication by α", that is, the map from L to itself sending x to αx. If L/K is inseparable
, then the trace map is identically 0.
When L/K is separable, a formula similar to the Galois case above can be obtained. If σ1, ..., σn are the distinct K-linear field embeddings of L into an algebraically closed field
containing K (where n is the degree of the extension L/K), then
When L/K is separable, the trace provides a duality theory via the trace form: the map from to K sending (x, y) to TrL/K(xy) is a nondegenerate, symmetric, bilinear form called the trace form. An example of where this is used is in algebraic number theory
in the theory of the different ideal
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the field trace is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
defined with respect to a finite field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
L/K. It is a K-linear map from L to K. As an example, if L/K is a Galois extension
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
and α is in L, then the trace of α is the sum of all the Galois conjugates of α, i.e.
where Gal(L/K) denotes the Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
of L/K.
For a general finite extension L/K, the trace of an element α can be defined as the trace
Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
of the K-linear map "multiplication by α", that is, the map from L to itself sending x to αx. If L/K is inseparable
Inseparable
Inseparable is another way of saying "cannot be separated".Inseparable may refer to:* in mathematics, inseparable differential equation, or in general the opposite of separable in any of its meanings...
, then the trace map is identically 0.
When L/K is separable, a formula similar to the Galois case above can be obtained. If σ1, ..., σn are the distinct K-linear field embeddings of L into an algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
containing K (where n is the degree of the extension L/K), then
Properties of the trace
As mentioned above, the trace is a K-linear map. Additionally, it behaves well in towers of fields: if M is a finite extension of L, then the trace from M to K is just the composition of the trace from M to L with the trace from L to K, i.e.When L/K is separable, the trace provides a duality theory via the trace form: the map from to K sending (x, y) to TrL/K(xy) is a nondegenerate, symmetric, bilinear form called the trace form. An example of where this is used is in algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
in the theory of the different ideal
Different ideal
In algebraic number theory, the different ideal is defined to account for the lack of duality in the ring of integers of an algebraic number field K, with respect to the field trace...
.