In
mathematics, the field trace is a particular
function defined with respect to a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb
Traditionally, a finite verb (from la, fīnītus, past partici ...
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
''L''/''K'', which is a
''K''-linear map from ''L'' onto ''K''.
Definition
Let ''K'' be a
field and ''L'' a finite extension (and hence an
algebraic extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field e ...
) of ''K''. ''L'' can be viewed as a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
over ''K''. Multiplication by ''α'', an element of ''L'',
:
,
is a ''K''-
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
of this vector space into itself. The ''trace'', Tr
''L''/''K''(''α''), is defined as the
trace (in the
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matric ...
sense) of this linear transformation.
For ''α'' in ''L'', let ''σ''(''α''), ..., ''σ''(''α'') be the
roots (counted with multiplicity) of the
minimal polynomial of ''α'' over ''K'' (in some extension field of ''K''). Then
:
If ''L''/''K'' is
separable then each root appears only once (however this does not mean the coefficient above is one; for example if ''α'' is the identity element 1 of ''K'' then the trace is
'L'':''K'' times 1).
More particularly, if ''L''/''K'' is a
Galois extension and ''α'' is in ''L'', then the trace of ''α'' is the sum of all the
Galois conjugate
In mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element , over a field extension , are the roots of the minimal polynomial of over . Conjugate elements are commonly called conju ...
s of ''α'',
i.e.,
:
where Gal(''L''/''K'') denotes the
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
of ''L''/''K''.
Example
Let
be a
quadratic extension of
. Then a
basis of
is
If
then the
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
of
is:
: