Primitive element theorem: Difference between revisions

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Revision as of 19:36, 6 July 2023

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For a non-separable extension <math>E/F</math> of [[characteristic p]], there is nevertheless a primitive element provided the degree [''E''&nbsp;:&nbsp;''F''] is ''p:'' indeed, there can be no non-trivial intermediate subfields since their degrees would be factors of the prime ''p''.

When [''E''&nbsp;:&nbsp;''F''] = ''p''², there may not be a primitive element (in which case there are infinitely many intermediate fields). The simplest example is <math>E=\mathbb{F}_p(T,U)</math>, the field of rational functions in two indeterminates ''T'' and ''U'' over the [[finite field]] with ''p'' elements, and <math>F=\mathbb{F}_p(T^p,U^p)</math>. In fact, for any &alpha; = ''g''(T,U) in ''<math>E'' \setminus F</math>, the [[Frobenius endomorphism]] shows that the element ''&alpha;''^''p'' lies in ''F'' , so &alpha; is a root of <math>f(X)=X^p-\alpha^p\in F[X]</math>, and &alpha; cannot be a primitive element (of degree ''p''² over ''F''), but instead ''F''(&alpha;) is a non-trivial intermediate field.

==Constructive results==