Field Trace
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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'', :m_\alpha:L\to L \text m_\alpha (x) = \alpha x, 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 :\operatorname_(\alpha) = :K(\alpha)sum_^n\sigma_j(\alpha). 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., :\operatorname_(\alpha)=\sum_\sigma(\alpha), 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 L = \mathbb(\sqrt) be a quadratic extension of \mathbb. Then a basis of L/\mathbb is \. If \alpha = a + b\sqrt 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 m_ is: :\left \begin a & bd \\ b & a \end \right /math>, and so, \operatorname_(\alpha) = :\mathbb(\alpha)left( \sigma_1(\alpha) + \sigma_2(\alpha)\right) = 1\times \left( \sigma_1(\alpha) + \overline(\alpha)\right) = a+b\sqrt + a-b\sqrt = 2a. The minimal polynomial of ''α'' is .


Properties of the trace

Several properties of the trace function hold for any finite extension. The trace is a ''K''- linear map (a ''K''-linear functional), that is :\operatorname_(\alpha a + \beta b) = \alpha \operatorname_(a)+ \beta \operatorname_(b) \text\alpha, \beta \in K. If then \operatorname_(\alpha) = :K\alpha. Additionally, trace 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. :\operatorname_=\operatorname_\circ\operatorname_.


Finite fields

Let ''L'' = GF(''q''''n'') be a finite extension of a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
''K'' = GF(''q''). Since ''L''/''K'' is a Galois extension, if ''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. :\operatorname_(\alpha)=\alpha + \alpha^q + \cdots + \alpha^. In this setting we have the additional properties: * \operatorname_(a^q) = \operatorname_(a) \text a \in L. * For any \alpha \in K, there are exactly q^ elements b\in L with \operatorname_(b) = \alpha. ''Theorem''. For ''b'' ∈ ''L'', let ''F''''b'' be the map a \mapsto \operatorname_(ba). Then if . Moreover, the ''K''-linear transformations from ''L'' to ''K'' are exactly the maps of the form ''F''''b'' as ''b'' varies over the field ''L''. When ''K'' is the
prime subfield In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive id ...
of ''L'', the trace is called the ''absolute trace'' and otherwise it is a ''relative trace''.


Application

A quadratic equation, with ''a'' ≠ 0, and coefficients in the finite field \operatorname(q) = \mathbb_q has either 0, 1 or 2 roots in GF(''q'') (and two roots, counted with multiplicity, in the quadratic extension GF(''q''2)). If the characteristic of GF(''q'') is odd, the discriminant indicates the number of roots in GF(''q'') and the classical quadratic formula gives the roots. However, when GF(''q'') has even characteristic (i.e., for some positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''h''), these formulas are no longer applicable. Consider the quadratic equation with coefficients in the finite field GF(2''h''). If ''b'' = 0 then this equation has the unique solution x = \sqrt in GF(''q''). If then the substitution converts the quadratic equation to the form: :y^2 + y + \delta = 0, \text \delta = \frac. This equation has two solutions in GF(''q'')
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...
the absolute trace \operatorname_(\delta) = 0. In this case, if ''y'' = ''s'' is one of the solutions, then ''y'' = ''s'' + 1 is the other. Let ''k'' be any element of GF(''q'') with \operatorname_(k) = 1. Then a solution to the equation is given by: : y = s = k \delta^2 + (k + k^2)\delta^4 + \ldots + (k + k^2 + \ldots + k^)\delta^. When ''h'' = 2''m + 1, a solution is given by the simpler expression: : y = s = \delta + \delta^ + \delta^ + \ldots + \delta^.


Trace form

When ''L''/''K'' is separable, the trace provides a duality theory via the trace form: the map from to ''K'' sending to Tr(''xy'') is a nondegenerate, symmetric bilinear form called the trace form. If ''L''/''K'' is a Galois extension, the trace form is invariant with respect to the Galois group. The trace form is used in algebraic number theory in the theory of the
different ideal In algebraic number theory, the different ideal (sometimes simply the different) is defined to measure the (possible) lack of duality in the ring of integers of an algebraic number field ''K'', with respect to the field trace. It then encodes the ...
. The trace form for a finite degree field extension ''L''/''K'' has non-negative
signature A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...
for any
field ordering In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field ...
of ''K''. The converse, that every Witt equivalence class with non-negative signature contains a trace form, is true for
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
s ''K''.Lorenz (2008) p.38 If ''L''/''K'' is an inseparable extension, then the trace form is identically 0. as footnoted in


See also

* Field norm *
Reduced trace In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' which is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple a ...


Notes


References

* * * * * * *


Further reading

* * Section VI.5 of {{DEFAULTSORT:Field Trace Field (mathematics)