The field F64 is the spitting field of x8 x over F4Solution

The field F_64 is the spitting field of x^8 - x over F_4.

Solution

It suffices to check that F₆₄ is the smallest field containing F₄ in which X⁸X has 8 roots. First, note that any element of F₆₄ is a root of X⁶⁴X=X(X⁶³1), and the latter factor is divisible by X⁷1 so that X⁸X indeed splits in F_64. On the other hand, the only field extension of F₄ smaller than F₆₄ is F₁₆, and F^×₁₆ cannot have an element of order 7 by Lagrange, meaning that X⁷1 cannot split. Therefore, F₆₄ is the smallest such field, and is by definition the splitting field.