(d) Let F be a field, R a nonzero ring and f: F→ R be a surjective homomorphism. Then prove that f is an isomorphism
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Q4d
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- Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.Prove that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]
- Let R be a commutative ring with unity. Prove that deg(f(x)g(x))degf(x)+degg(x) for all nonzero f(x), g(x) in R[ x ], even if R in not an integral domain.Suppose that f(x),g(x), and h(x) are polynomials over the field F, each of which has positive degree, and that f(x)=g(x)h(x). Prove that the zeros of f(x) in F consist of the zeros of g(x) in F together with the zeros of h(x) in F.If is a finite field with elements, and is a polynomial of positive degree over , find a formula for the number of elements in the ring .
- Let :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove that is a ring homomorphism. Is (e)=e where e is the unity in 3 and e is the unity in 12?Prove Theorem If and are relatively prime polynomials over the field and if in , then in .Let where is a field and let . Prove that if is irreducible over , then is irreducible over .