MAT 535 Notes on The Fundamental Theorem of Galois Theory

Let F be a commutative ring with 1. For every set Σ, denote by F the set of all set functions a : Σ → F . This is an F -algebra under pointwise addition, pointwise scaling, and pointwise multiplication; i.e., for a, b in F and for λ in F , for every σ in Σ we define (a+ b)(σ) := a(σ) + b(σ), (λ · b)(σ) := λ · b(σ), (a · b)(σ) := a(σ) · b(σ). These structures make F into a commutative F -algebra whose additive identity is the constant function with value 0 and whose multiplicative identity is the constant function with value 1. For every element λ in F , we will denote by λ the constant function in F with value λ.