primitive element theorem
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In field theory, the primitive element theorem is a result characterizing the finite degree field extensions that can be generated by a single element. Such a generating element is called a primitive element of the field extension, and the extension is called a
simple extension In field theory, a simple extension is a field extension which is generated by the adjunction of a single element. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization o ...
in this case. The theorem states that a finite extension is simple if and only if there are only finitely many intermediate fields. An older result, also often called "primitive element theorem", states that every finite separable extension is simple; it can be seen as a consequence of the former theorem. These theorems imply in particular that all
algebraic number fields 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 fi ...
over the rational numbers, and all extensions in which both fields are finite, are simple.


Terminology

Let E/F be a '' field extension''. An element \alpha\in E is a ''primitive element'' for E/F if E=F(\alpha), i.e. if every element of E can be written as a
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
in \alpha with coefficients in F. If there exists such a primitive element, then E/F is referred to as a ''
simple extension In field theory, a simple extension is a field extension which is generated by the adjunction of a single element. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization o ...
''. If the field extension E/F has primitive element \alpha and is of finite degree n = :F/math>, then every element ''x'' of ''E'' can be written uniquely in the form :x=f_^+\cdots+f_1+f_0, where f_i\in F for all ''i''. That is, the set :\ is a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
for ''E'' 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 can ...
over ''F''.


Example

If one adjoins to the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s F = \mathbb the two irrational numbers \sqrt and \sqrt to get the extension field E=\mathbb(\sqrt,\sqrt) of
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
4, one can show this extension is simple, meaning E=\mathbb(\alpha) for a single \alpha\in E. Taking \alpha = \sqrt + \sqrt , the powers 1, α , α2, α3 can be expanded as linear combinations of 1, \sqrt, \sqrt, \sqrt with integer coefficients. One can solve this system of linear equations for \sqrt and \sqrt over \mathbb(\alpha), to obtain \sqrt = \tfrac12(\alpha^3-9\alpha) and \sqrt = -\tfrac12(\alpha^3-11\alpha). This shows α is indeed a primitive element: :\mathbb(\sqrt 2, \sqrt 3)=\mathbb(\sqrt2 + \sqrt3).


The theorems

The classical primitive element theorem states: :Every separable field extension of finite degree is simple. This theorem applies to algebraic number fields, i.e. finite extensions of the rational numbers Q, since Q has characteristic 0 and therefore every finite extension over Q is separable. The following primitive element theorem (
Ernst Steinitz Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician. Biography Steinitz was born in Laurahütte (Siemianowice Śląskie), Silesia, Germany (now in Poland), the son of Sigismund Steinitz, a Jewish coal merchant, and ...
) is more general: :A finite field extension E/F is simple if and only if there exist only finitely many intermediate fields ''K'' with E\supseteq K\supseteq F. Using the
fundamental theorem of Galois theory In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory. In its most basi ...
, the former theorem immediately follows from the latter.


Characteristic ''p''

For a non-separable extension E/F of
characteristic p 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 ide ...
, there is nevertheless a primitive element provided the degree 'E'' : ''F''is ''p:'' indeed, there can be no non-trivial intermediate subfields since their degrees would be factors of the prime ''p''. When 'E'' : ''F''= ''p''2, there may not be a primitive element (in which case there are infinitely many intermediate fields). The simplest example is E=\mathbb_p(T,U), the field of rational functions in two indeterminates ''T'' and ''U'' over the
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, subtr ...
with ''p'' elements, and F=\mathbb_p(T^p,U^p). In fact, for any α = ''g''(T,U) in ''E'', the
Frobenius endomorphism In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphis ...
shows that the element ''α''''p'' lies in ''F'' , so α is a root of f(X)=X^p-\alpha^p\in F /math>, and α cannot be a primitive element (of degree ''p''2 over ''F''), but instead ''F''(α) is a non-trivial intermediate field.


Constructive results

Generally, the set of all primitive elements for a finite separable extension ''E'' / ''F'' is the complement of a finite collection of proper ''F''-subspaces of ''E'', namely the intermediate fields. This statement says nothing in the case of
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, subtr ...
s, for which there is a computational theory dedicated to finding a generator of the
multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
of the field (a
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
), which is ''a fortiori'' a primitive element (see
primitive element (finite field) In field theory, a primitive element of a finite field is a generator of the multiplicative group of the field. In other words, is called a primitive element if it is a primitive th root of unity in ; this means that each non-zero element of ...
). Where ''F'' is infinite, a
pigeonhole principle In mathematics, the pigeonhole principle states that if items are put into containers, with , then at least one container must contain more than one item. For example, if one has three gloves (and none is ambidextrous/reversible), then there mu ...
proof technique considers the linear subspace generated by two elements and proves that there are only finitely many linear combinations :\gamma = \alpha + c \beta\ with ''c'' in ''F'', that fail to generate the subfield containing both elements: :as F(\alpha,\beta)/F(\alpha+c\beta) is a separable extension, if F(\alpha+c\beta) \subsetneq F(\alpha,\beta) there exists a non-trivial embedding \sigma : F(\alpha,\beta)\to \overline whose restriction to F(\alpha+c\beta) is the identity which means \sigma(\alpha)+c \sigma(\beta) = \alpha+c \beta and \sigma(\beta) \ne \beta so that c = \frac. This expression for ''c'' can take only (\alpha):F (\beta):F/math> different values. For all other value of c\in F then F(\alpha,\beta) = F(\alpha+c\beta). This is almost immediate as a way of showing how Steinitz' result implies the classical result, and a bound for the number of exceptional ''c'' in terms of the number of intermediate fields results (this number being something that can be bounded itself by Galois theory and ''a priori''). Therefore, in this case trial-and-error is a possible practical method to find primitive elements.


History

In his First Memoir of 1831,
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
sketched a proof of the classical primitive element theorem in the case of a
splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a poly ...
of a polynomial over the rational numbers. The gaps in his sketch could easily be filled (as remarked by the referee
Siméon Denis Poisson Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electri ...
; Galois' Memoir was not published until 1846) by exploiting a theorem of Joseph-Louis Lagrange from 1771, which Galois certainly knew. It is likely that Lagrange had already been aware of the primitive element theorem for splitting fields. Galois then used this theorem heavily in his development of 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 ...
. Since then it has been used in the development of
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
and the
fundamental theorem of Galois theory In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory. In its most basi ...
. The two primitive element theorems were proved in their modern form by Ernst Steinitz, in an influential article on field theory in 1910; Steinitz called the "classical" one ''Theorem of the primitive elements'' and the other one ''Theorem of the intermediate fields''. Emil Artin reformulated Galois theory in the 1930s without the use of the primitive element theorems.


References

{{Reflist


External links


J. Milne's course notes on fields and Galois theory



The primitive element theorem at planetmath.org

The primitive element theorem on Ken Brown's website (pdf file)
Field (mathematics) Theorems in abstract algebra