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By David B Surowski

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I) φ : S → F is injective. (ii) F = φ(S) . (iii) Via the map φ : {s} → Z, φ(s) = 1, the additive group Z is free on one generator. (iv) If F is free on a set with more than one generator, then F is nonabelian. The following is absolutely fundamental. 2 If S is nonempty, then a free group exists on S, and is unique up to isomorphism. Now let G be an arbitrary group. It is clear that G is the homomorphic image of some free group F . Indeed, let F be the free group on the set G; the map F → G is then that induced by 1G : G → G.

1. BASICS 45 6. Let F = C(x), where C is the complex number field and x is an indeterminate. Assume that F ⊆ K and that K contains an element y such that y 2 = x(x − 1). , F(y) is a “simple transcendental extension” of C. 7. Let F ⊆ K be a field extension. If the subfields of K containing F are totally ordered by inclusion, prove that K is a simple extension of F. ) 8. Let Q ⊆ K be a field extension. , if α ∈ K, then α ∈ K. Prove that [K : Q] = ∞. ) 9. Let F be a field, contained as a subring of the integral domain R.

The reader should also be familiar with the concept of the characteristic of a field. If F and K are fields with F ⊆ K, we say that K is an extension of F. Of fundamental importance here is the observation that if F ⊆ K is an extension of fields, then K can be regarded as a vector space over F. It is customary to call the F-dimension of K the degree of K over F, and to denote this degree by [K : F]. The following simple result is fundamental. 1 Let F ⊆ E ⊆ K be an extension of fields. Then [K : F] < ∞ if and only if each of [K : E], [E : F] < ∞, in which case [K : F] = [K : E] · [E : F].

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A workbook in higher algebra by David B Surowski

by David

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