Prove that if F is a field and f(x) is in F[x]\F, then dimF(F[x]/<f(x)>) = deg(f(x)).
Prove that if F is a field and f(x) is in F[x]\F, then dimF(F[x]/<f(x)>) = deg(f(x)).
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...
14. If f(a) and g(x) are polynomials over the field F, and h(x)-f(x) t gx), prove that h(c)-f(c) + g(c) for all c in F. 15. If f(x) and g(x) are polynomials over the field F, and p(x)fx)g(x), prove that p(c) -f(c)g(c) for all c in F
- Let F be a field. Prove: For all (o), 9(a), (x) € F2, if f(x) and (w) are relatively prime and (a)/(x), then (a) and f(a) are relatively prime.
10. Suppose that f(x) E Z[x] and f(x) is irreducible over Z, where p is a prime. If deg f(x) n, prove that 2,[x]/f(x)) is a field with p" elements. 11. Construct a field of order 25.
Suppose that x is a tree such that for every vertex v of x, (deg(v))%3=1. Prove that x cannot have 25 vertices.
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...
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).
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]
NAME STUDENT NO: DEPARTMENT Q-2) Let F be a field and h Fr], deg h > 0. Show that the map L: FxFz] given by f (h) is a linear operator. Find Ker L. Show that L is an isomorphism if and only if deg h 1. Solution:
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...