This paper deals with
the classical Galois theory in the context of the Arens-Hoffman extension
B = A[x]∕(α(x)) of a commutative Banach algebra A (with identity over
the complex field 𝒞) with respect to a monic polynomial α(x) over A with
an invertible discriminant. We show that the fundamental theorem of the
Galois theory for commutative rings [S.U. Chase, D.K. Harrison, and A.
Rosenberg, Galois theory and cohomology of commutative rings, Memoirs, Amer.
Math. Soc. No. 52 (1965)] applies to our situation. The fixed algebras of the
subgroups of the Galois group are then characterized for the case where A is
semi-simple. The techniques are primarily topological and consist in examining
the relationships between ΦB and ΦA, where the Φ’s denote the respective
carrier spaces of the Banach algebras A and B together with the usual weak
∗ topology.
