In field theory, the primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element.
This theorem implies in particular that all algebraic number fields over the rational numbers, and all extensions in which both fields are finite, are simple.
can be written as a rational function in
is referred to as a simple extension.
can be written in the form for unique coefficients
That is, the set is a basis for E as a vector space over F. The degree n is equal to the degree of the irreducible polynomial of α over F, the unique monic
of minimal degree with α as a root (a linear dependency of
, and these extend to automorphisms of L in the Galois group,
of degree 4, one can show this extension is simple, meaning
, the powers 1, α, α2, α3 can be expanded as linear combinations of 1,
One can solve this system of linear equations for
This shows that α is indeed a primitive element: One may also use the following more general argument.
The primitive element theorem states: 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.
of characteristic p, 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] = p2, there may not be a primitive element (in which case there are infinitely many intermediate fields by Steinitz's theorem).
, the Frobenius endomorphism shows that the element
lies in F , so α is a root of
, and α cannot be a primitive element (of degree p2 over F), but instead F(α) is a non-trivial intermediate field.
By induction, it suffices to prove that any finite extension
α = β + c γ
β = α − c γ ∈
fail to give a primitive element
α = β + c γ
is finite, we simply take
to be a primitive root of the finite extension field
In his First Memoir of 1831, published in 1846,[2] Évariste Galois sketched a proof of the classical primitive element theorem in the case of a splitting field of a polynomial over the rational numbers.
The gaps in his sketch could easily be filled[3] (as remarked by the referee Poisson) by exploiting a theorem[4][5] of Lagrange from 1771, which Galois certainly knew.
It is likely that Lagrange had already been aware of the primitive element theorem for splitting fields.
Since then it has been used in the development of Galois theory and the fundamental theorem of Galois theory.
The primitive element theorem was proved in its modern form by Ernst Steinitz, in an influential article on field theory in 1910, which also contains Steinitz's theorem;[6] Steinitz called the "classical" result Theorem of the primitive elements and his modern version Theorem of the intermediate fields.
Emil Artin reformulated Galois theory in the 1930s without relying on primitive elements.