Primitive element theorem: Difference between revisions

Revision as of 21:49, 19 December 2023

Revision as of 22:01, 19 December 2023

Line 4:

Let <math>E/F</math> be a ''[[field extension]]''. An element <math>\alpha\in E</math> is a ''primitive element'' for <math>E/F</math> if <math>E=F(\alpha),</math> i.e. if every element of <math>E</math> can be written as a [[rational function]] in <math>\alpha</math> with coefficients in <math>F</math>. If there exists such a primitive element, then <math>E/F</math> is referred to as a ''[[simple extension]]''.

If the field extension <math>E/F</math> has primitive element <math>\alpha</math> and is of finite [[Degree of a field extension|degree]] <math>n = [E:F]</math>, then every element ''γ''<math>\gamma\in of ''E''</math> can be written in the form

:<math>\gamma =a_0+a_1{\alpha}+\cdots+a_{n-1}{\alpha}^{n-1}, </math>

Line 12:

:<math>\{1,\alpha,\ldots,{\alpha}^{n-1}\}</math>

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(xX)\in F[xX] </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''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 \supset E</math>.

== Example ==