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.

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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