Suppose is a finite group of automorphisms of a field . Form the ring which is the subring of the ring of endomorphisms of generated by and multiplication by elements of . Note that if then the composite equals the composite . Show that the -subalgebras of the group ring are those of the form for a subgroup of .

*Note* By an -subalgebra we mean those subrings that contain all the multiplications by elements .

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Consider such a subring , and note that is a submonoid of , and since is finite, it is a subgroup. Evidently by the condition that be an -subalgebra. Assuming this is a proper inclusion, pick an element of the form such that no lies in and each is nonzero, with minimal. Note that since else . Now choose such that and note that lies in , and subtracting from this gives a contradiction by the minimality of . This is the same trick one uses to show Dedekind’s theorem that characters of monoid onto fields are linearly independent!

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