Simple extension

In field theory, a simple extension is a field extension which is generated by the adjunction of a single 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

The element θ is called a primitive element, or generating element, for the extension; we also say that L is generated over K by θ.

Every finite field is a simple extension of the prime field of the same characteristic. More precisely, if p is a prime number and ${\displaystyle q=p^{d}}$ the field ${\displaystyle \mathbb {F} _{q}}$ of q elements is a simple extension of degree d of ${\displaystyle \mathbb {F} _{p}.}$ This means that it is generated by an element θ that is a root of an irreducible polynomial of degree d. However, in this case, θ is normally not referred to as a primitive element.

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