Primitive element theorem
Encyclopedia
In mathematics
, more specifically in the area of modern algebra known as field theory
, the primitive element theorem or Artin's theorem on primitive elements is a result characterizing the finite degree field extension
s that possess a primitive element. More specifically, the primitive element theorem characterizes those finite degree extensions
such that there exists 
with 
.
be an arbitrary field extension. An element 
is said to be a primitive element for 
when
In this situation, the extension
is referred to as a simple extension
. Then every element x of E can be written in the form
where 
for all i, and
is fixed. That is, if 
is separable of degree n, there exists 
such that the set
is a basis for E as a vector space
over F.
For instance, the extensions
and 
are simple extensions with primitive elements 
and x, respectively (
denotes the field of rational functions in the indeterminate x over 
).
, around 1930. From the time of Galois, the role of primitive elements had been to represent a splitting field
as generated by a single element. This (arbitrary) choice of such an element was bypassed in Artin's treatment. At the same time, considerations of construction of such an element receded: the theorem becomes an existence theorem
.
The following theorem of Artin then takes the place of the classical primitive element theorem.
Theorem
Let
be a finite degree field extension. Then 
for some element 
if and only if there exist only finitely many intermediate fields K with 
.
A corollary to the theorem is then the primitive element theorem in the more traditional sense (where separability was usually tacitly assumed):
Corollary
Let
be a finite degree separable extension
. Then
for some 
.
The corollary applies to algebraic number field
s, i.e. finite extensions of the rational numbers Q, since Q has characteristic
0 and therefore every extension over Q is separable.
the field of rational functions in two indeterminates T and U over the finite field
with p elements, and L is obtained from K by adjoining a p-th root of T, and of U. In fact one can see that for any α in L, the element αp lies in K, but a primitive element must have degree p2 over K.
s, for which there is a computational theory dedicated to finding a generator of the multiplicative group
of the field (a cyclic group
), which is a fortiori a primitive element. Where K is infinite, a pigeonhole principle proof technique considers the linear subspace generated by two elements and proves that there are only finitely many linear combinations
with c in K in it, that fail to generate the subfield containing both elements. This is almost immediate as a way of showing how Artin's 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. See the Example.
s roots of both polynomial
s
and
say α and β respectively, to get a field K = Q(α, β) of degree
4 over Q, that the extension is simple and there exists a primitive element γ in K so that K = Q(γ). One can in fact check that with
the powers γ i for 0 ≤ i ≤ 3 can be written out as linear combination
s of 1, α, β and αβ with integer coefficients. Taking these as a system of linear equations, or by factoring, one can solve for α and β over Q(γ) (one gets, for instance, α=
), which implies that this choice of γ is indeed a primitive element in this example. A simpler argument, assuming the knowledge of all the subfields as given by Galois theory, is to note the independence of 1, α, β and αβ over the rationals; this shows that the subfield generated by γ cannot be that generated α or β, nor in fact that generated by αβ, exhausting all the subfields of degree 2. Therefore it must be the whole field.
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...
, more specifically in the area of modern algebra known as field theory
Field theory (mathematics)
Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....
, the primitive element theorem or Artin's theorem on primitive elements is a result characterizing the finite degree 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...
s that possess a primitive element. More specifically, the primitive element theorem characterizes those finite degree extensions
such that there exists with .Terminology
Let be an arbitrary field extension. An element is said to be a primitive element for whenIn this situation, the extension
is referred to as a simple extensionSimple extension
In mathematics, more specifically in field theory, a simple extension is a field extension which is generated by the adjunction of a single element...
. Then every element x of E can be written in the form
where for all i, and
is fixed. That is, if is separable of degree n, there exists such that the setis a basis for E as a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over F.
For instance, the extensions
and are simple extensions with primitive elements and x, respectively ( denotes the field of rational functions in the indeterminate x over ).Existence statement
The interpretation of the theorem changed with the formulation of the theory of Emil ArtinEmil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
, around 1930. From the time of Galois, the role of primitive elements had been to represent a splitting field
Splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors into linear factors.-Definition:...
as generated by a single element. This (arbitrary) choice of such an element was bypassed in Artin's treatment. At the same time, considerations of construction of such an element receded: the theorem becomes an existence theorem
Existence theorem
In mathematics, an existence theorem is a theorem with a statement beginning 'there exist ..', or more generally 'for all x, y, ... there exist ...'. That is, in more formal terms of symbolic logic, it is a theorem with a statement involving the existential quantifier. Many such theorems will not...
.
The following theorem of Artin then takes the place of the classical primitive element theorem.
Theorem
Let
be a finite degree field extension. Then for some element if and only if there exist only finitely many intermediate fields K with .A corollary to the theorem is then the primitive element theorem in the more traditional sense (where separability was usually tacitly assumed):
Corollary
Let
be a finite degree separable extensionSeparable extension
In modern algebra, an algebraic field extension E\supseteq F is a separable extension if and only if for every \alpha\in E, the minimal polynomial of \alpha over F is a separable polynomial . Otherwise, the extension is called inseparable...
. Then
for some .The corollary applies to algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
s, i.e. finite extensions of the rational numbers Q, since Q has characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
0 and therefore every extension over Q is separable.
Counterexamples
For non-separable extensions, necessarily in characteristic p with p a prime number, then at least when the degree [L : K] is p, then L / K has a primitive element, because there are no intermediate subfields. When [L : K] = p2 then there may not be a primitive element (and therefore there are infinitely many intermediate fields). This happens, for example if K is- Fp(T, U),
the field of rational functions in two indeterminates T and U over the finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
with p elements, and L is obtained from K by adjoining a p-th root of T, and of U. In fact one can see that for any α in L, the element αp lies in K, but a primitive element must have degree p2 over K.
Constructive results
Generally, the set of all primitive elements for a finite separable extension L / K is the complement of a finite collection of proper K-subspaces of L, namely the intermediate fields. This statement says nothing for the case of finite fieldFinite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s, for which there is a computational theory dedicated to finding a generator of the multiplicative group
Multiplicative group
In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context*any group \scriptstyle\mathfrak \,\! whose binary operation is written in multiplicative notation ,*the underlying group under multiplication of the invertible elements of...
of the field (a cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
), which is a fortiori a primitive element. Where K is infinite, a pigeonhole principle proof technique considers the linear subspace generated by two elements and proves that there are only finitely many linear combinations
with c in K in it, that fail to generate the subfield containing both elements. This is almost immediate as a way of showing how Artin's 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. See the Example.
Example
It is not, for example, immediately obvious that if one adjoins to the field Q of rational numberRational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s roots of both polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s
and
say α and β respectively, to get a field K = Q(α, β) of degree
Degree of a field extension
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently.-...
4 over Q, that the extension is simple and there exists a primitive element γ in K so that K = Q(γ). One can in fact check that with
the powers γ i for 0 ≤ i ≤ 3 can be written out as linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
s of 1, α, β and αβ with integer coefficients. Taking these as a system of linear equations, or by factoring, one can solve for α and β over Q(γ) (one gets, for instance, α=
), which implies that this choice of γ is indeed a primitive element in this example. A simpler argument, assuming the knowledge of all the subfields as given by Galois theory, is to note the independence of 1, α, β and αβ over the rationals; this shows that the subfield generated by γ cannot be that generated α or β, nor in fact that generated by αβ, exhausting all the subfields of degree 2. Therefore it must be the whole field.