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In [[field theory (mathematics)|field theory]], the '''primitive element theorem''' states that every [[degree of a field extension|finite]] [[separable extension|separable]] [[field extension]] is [[Simple extension|simple]], i.e. generated by a single element. This theorem implies in particular that all [[Algebraic number field|algebraic number fields]] over the rational numbers, and all extensions in which both fields are finite, are simple.
== Terminology ==
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is a [[Basis (linear algebra)|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 <math>f(X)\in F[X] </math> of minimal degree with ''α'' as a root (a linear dependency of <math>\{1,\alpha,\ldots,\alpha^{n-1},\alpha^n\} </math>).
If ''L'' is a [[splitting field]] of <math>f(X)</math> containing its ''n'' distinct roots <math>\alpha_1,\ldots,\alpha_n </math>, then there are ''n'' [[Homomorphism|field embeddings]] <math>\sigma_i : F(\alpha)\hookrightarrow L </math> defined by <math>\sigma_i(\alpha)=\alpha_i </math> and <math>\sigma(a)=a </math> for <math>a\in F </math>, and these extend to automorphisms of ''L'' in the [[Galois group]], <math>\sigma_1,\ldots,\sigma_n\in \mathrm{Gal}(L/F) </math>. Indeed, for an extension field with <math>[E: F]=n </math>, an element <math>\alpha</math> is a primitive element if and only if <math>\alpha</math> has ''n'' distinct conjugates <math>\sigma_1(\alpha),\ldots,\sigma_n(\alpha)</math> in some splitting field <math>L \
== Example ==
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