4. Suppose K is a finite field, and let : (Z.+.0) → (K. +.0k) be the unique group homomorphism with (1) = 1k. a) Show that is a ring homomorphism. b) Show that ker=pZ for some prime p. p is called the characteristic of the field K. c) Show that induces an injective ring homomorphism : F→ K. d) Deduce that K is a vector space over Fp. e) Conclude that K-p" for some n.
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- Prove that if R is a field, then R has no nontrivial ideals.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.)Since this section presents a method for constructing a field of quotients for an arbitrary integral domain D, we might ask what happens if D is already a field. As an example, consider the situation when D=5. a. With D=5, write out all the elements of S, sort these elements according to the relation , and then list all the distinct elements of Q. b. Exhibit an isomorphism from D to Q.
- Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.Use Theorem to show that each of the following polynomials is irreducible over the field of rational numbers. Theorem Irreducibility of in Suppose is a polynomial of positive degree with integral coefficients and is a prime integer that does not divide. Let Where for If is irreducible in then is irreducible in .
- a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].. a. Let, and . Show that and are only ideals of and hence is a maximal ideal. b. Show that is not a field. Hence Theorem is not true if the condition that is commutative is removed. Theorem 6.22 Quotient Rings That are Fields. Let be a commutative ring with unity, and let be an ideal of . Then is a field if and only if is a maximal ideal of .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.