Galois correspondence via group algebras

Suppose G is a finite group of automorphisms of a field E. Form the ring E(G) which is the subring of the ring of endomorphisms of E generated by G and multiplication by elements of E. Note that if \lambda\in E then the composite g\cdot \mu_\lambda equals the composite \mu_{g(\lambda)} \cdot g. Show that the E-subalgebras of the group ring E(G) are those of the form E(H) for H a subgroup of G.

Note By an E-subalgebra we mean those subrings that contain all the multiplications by elements \lambda \in E.

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Galois correspondence via group algebras

One thought on “Galois correspondence via group algebras

  1. Consider such a subring S, and note that G\cap S=H is a submonoid of G, and since G is finite, it is a subgroup. Evidently E(H)\subseteq S by the condition that S be an E-subalgebra. Assuming this is a proper inclusion, pick an element of the form a_1s_1+\cdots+a_rs_r such that no s_i lies in H and each a_j is nonzero, with r minimal. Note that r>1 since else a_1s_1\in S\cap G=H. Now choose x such that s_1(x)\neq s_2(x) and note that \sum a_i s_i\cdot x=\sum a_i s_i(x) s_i lies in S, and subtracting s_1(x)\sum a_is_i from this gives a contradiction by the minimality of r. 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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