# Abstract Algebra (Course 311) by Wilkins D.R.

By Wilkins D.R.

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A K-homomorphism θ: L → M is a homomorphism of fields which satisfies θ(a) = a for all a ∈ K. A K-monomorphism is an injective K-homomorphism. A K-isomorphism is a bijective K-homomorphism. A K-automorphism of L is a K-isomorphism mapping L onto itself. Two extensions L1 : K and L2 : K of a field K are said to be K-isomorphic (or isomorphic) if there exists a K-isomorphism ϕ: L1 → L2 between L1 and L2 . If L: K is a field extension then we can regard L as a vector space over the field K. If L is a finite-dimensional vector space over K then we say that the extension L: K is finite.

We denote K({α1 , α2 , . . , αk }) by K(α1 , α2 , . . , αk ) for any finite subset {α1 , α2 , . . , αk } of L. In particular K(α) denotes the field obtained by adjoining some element α of L to K. A field extension L: K is said to be simple if there exists some element α of L such that L = K(α). 2 Algebraic Field Extensions Definition Let L: K be a field extension, and let α be an element of L. If there exists some non-zero polynomial f ∈ K[x] with coefficients in K such that f (α) = 0, then α is said to be algebraic over K; otherwise α is said to be transcendental over K.

Two rings are said to be isomorphic if there is an isomorphism between them. 34 The verification of the following result is a straightforward exercise. 7 Let ϕ: R → S be a homomorphism from a ring R to a ring S, and let I be an ideal of R satisfying I ⊂ ker ϕ. Then there exists a unique homomorphism ϕ: R/I → S such that ϕ(I + x) = ϕ(x) for all x ∈ R. Moreover ϕ: R/I → S is injective if and only if I = ker ϕ. 8 Let ϕ: R → S be ring homomorphism. Then ϕ(R) is isomorphic to R/ ker ϕ. 4 The Characteristic of a Ring Let R be a ring, and let r ∈ R.