5. Let E be a normal extension of Q, and let Fi, F CE be two normal subex- tensionsnAssume also that Fin F2 Prove that Gal( E/O) Q and that Fi and F2 generate E is a field isomorphic to the product Gal(F/Q) x Gal(F2/Q) is X 5. Let E be a normal extension of Q, and let Fi, F CE be two normal subex- tensionsnAssume also that Fin F2 Prove that Gal( E/O) Q and that Fi and F2...
5. (a) (5 points) Let R F[x] for a field F. Let f, g E R be nonzero. Prove that (f(x)) = (g(x)) if and only if g(x) = af(x) for some constant a E F. (b) (5 points) Let R be any ring. Prove that the nilradical Vo is contained in the intersection of all prime ideals.
(7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X, define d(f) = f2. : X → X is differentiable, and Prove that φ find φ'(f). (b) Given f e X, define 9(f) = J0 [f(t)]2dt. Prove that Ψ : X → R is differentiable. and find Ψ(f). (7) In this problem let X denote the vector space C(0, 1) with the sup norm. (a) Given f e X,...
4. Let F be a field. Prove that for all polynonials f(x), g(x), h (z) є FI2], if f(x) divides g(x) and f(z) divides h(r), then for all polynomials s(r),t() E Fr, f() divides s()g(r) +t(x)h(r). 4. Let F be a field. Prove that for all polynonials f(x), g(x), h (z) є FI2], if f(x) divides g(x) and f(z) divides h(r), then for all polynomials s(r),t() E Fr, f() divides s()g(r) +t(x)h(r).
Problem 4. Let K be a field and let f ∈ K[x]. Show that if 1+f2 has a factor of odd degree in K[x] then there is an a ∈ K such that a2 = −1.
1. Let F be a field and let F(X) be the field of rational functions ), with coefficients in F. Let K be any field such that F C KCFX and K F. Prove that F(X) : K] < oo. 1. Let F be a field and let F(X) be the field of rational functions ), with coefficients in F. Let K be any field such that F C KCFX and K F. Prove that F(X) : K]
6. Let F be a field and a Fx] a nonconstant polynomial. Denote (that is, (a(x)) is the set of all polynomials in Flr] which are divisble by a()). Then (a) Prove that (a(x)) is a subgroup of the abelian group (Flx],. (b) consider the operation on F[r]/(a()) given by Prove that this operation is well-defined. (c) Prove that the quotient F]/(a(x) is a commutative ing with identity (d) What happens if the polynoial a() is constant? 6. Let F...
2. (D5) Let n = o(a) and assume that a =bk. Prove that <a >=<b> if and only if n and k are relatively prime.
please help me,thanks! 3. Let Fo be a field with 9 elements. Consider the set S () e Fo] deg(f()) 18, f( f(1) (2)) (4) 0 and (a) Compute IS. (b) Prove that S is a vector space over F (c) Compute dimF, S Let V be a vector space over F. Prove that X C V is a subspace if and only if v, w E X implies av+wEX for every aEF 3. Let Fo be a field with...
Please write legibly and show all work! The goal is to prove the product rule for polynomials over a field F. Let f(x),g(x) E Fx. Prove that d )g))g) This will be done in three steps. (a) Show it is true when fx)s) are monomials f(x)-a,stx) (b) Show it is true when f(x) -as any polynomial but g(x) bx is a i-0 monomial Use your result from (a) and the proat (x)g) 1n (c) Show it is true in the...