On the Number of Generators of a Separable Algebra over a Finite Field
نویسنده
چکیده
Let F be a field and let E be an étale algebra over F , that is, a finite product of finite separable field extensions, E = F1 × · · · × Fr. The classical primitive element theorem asserts that if r = 1, then E is generated by one element as an F -algebra. The same is true for any r > 1, provided that F is infinite. However, if F is a finite field and r > 2, the primitive element theorem fails in general. In this paper we give a formula for the minimal number of generators of E when F is finite. We also obtain upper and lower bounds on the number of generators of a (not necessarily commutative) separable algebra over a finite field.
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