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Primitive element theorem

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 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 over the rational numbers, and all extensions in which both fields are finite, are simple. Let be a field extension. An element is a primitive element for if i.e. if every element of can be written as a rational function in with coefficients in . If there exists such a primitive element, then is referred to as a simple extension. If the field extension has primitive element and is of finite degree , then every element x of E can be written uniquely in the form where for all i. That is, the set is a basis for E as a vector space over F. If one adjoins to the rational numbers the two irrational numbers and to get the extension field of degree 4, one can show this extension is simple, meaning for a single . Taking , the powers 1, α, α2, α3 can be expanded as linear combinations of 1, , , with integer coefficients. One can solve this system of linear equations for and over , to obtain and . This shows that α is indeed a primitive element: 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) is more general: A finite field extension is simple if and only if there exist only finitely many intermediate fields K with .

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Related concepts (9)
Algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic field extension). Thus is a field that contains and has finite dimension when considered as a vector space over . The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
Glossary of field theory
Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.) A field is a commutative ring (F,+,*) in which 0≠1 and every nonzero element has a multiplicative inverse. In a field we thus can perform the operations addition, subtraction, multiplication, and division. The non-zero elements of a field F form an abelian group under multiplication; this group is typically denoted by F×; The ring of polynomials in the variable x with coefficients in F is denoted by F[x].
Simple extension
In field theory, a simple extension is a field extension which is generated by the adjunction of a single element, called a primitive element. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization of the finite simple extensions. A field extension L/K is called a simple extension if there exists an element θ in L with This means that every element of L can be expressed as a rational fraction in θ, with coefficients in K; that is, it is produced from θ and elements of K by the field operations +, −, •, / .
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