1. Let F be a field and let F(X) be the field of rational functions ), with coefficients in F. Le...
11-3. Let K Ze(t) be the field of rational functions, let f(x) = xP - 2 - t€ K[2], and let E/K be a splitting field of f(x). Prove that Gal(E/K) 2 2p but that f(x) is not solvable by radicals.
Let f(x) = ax^2 +bx +c be a quadratic whose coefficients a, b, c are rational. Prove that if f(x) has one rational root, then the other root is also rational.
let f(x) and g(x) be two polynomials with rational coefficients. Let d(x) be the greatest common of f(x) and g(x) in Q[x] (Q as in the set of rational numbers) and e(x) the greatest common divisor of f(x) and g(x) in C[x] (C and in set of complex numbers). is d(x) = e(x)
3. Let f, g : a, bl → R be functions such that f is integrable, g is continuous. and g(x) >0 for al x E [a, b]. Since both f,g are bounded, let K> 0 be such that f(x)| 〈 K and g(x)-K for all x E la,b] (a) Let η 〉 0 be given. Prove that there is a partition P of a,b] such that for all i (b) Let P be a partition as in (a). Prove...
3. Let f, g : a, b] → R be functions such that f is integrable, g is continuous. and g(x) 〉 0 for all x є a,b]. Since both f, g are bounded, let K 〉 0 be such that |f(x) K and g(x) < K for all x E [a,b (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that for all i 2. (b) Let P be a...
3. Let f, g : [a,b] → R be functions such that f is integrable, g is continuous, and g(x) >0 for all r E [a, b] Since both f,g are bounded, let K >0 be such that lf(z)| K and g(x) K for all x E [a3] (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that U (P. f) _ L(P./) < η and Mi(P4)-mi(P4) < η for all...
1. Let Q be the set of polynomials with rational coefficients. You may assume that this is an abelian group under addition. Consider the function Ql] Q[x] given by p(px)) = p'(x), where we are taking the derivative. Show that is a group homomorphism. Determine the kernel of 2. Let G and H be groups. Show that (G x H)/G is isomorphic to H. Hint: consider defining a surjective homomorphism p : Gx HH with kernel G. Then apply the...
1-> X- Let f :S → R and g:S → R be functions and c be a cluster point. Assume lim f (x), lim g(x) exists. Using the definition of the limit prove the following lim(af (x) + Bg(x)) =a lim f(x) + Blim g(x) for any a,ßeR xc XC X-> b. lim( f(x))} = (lim f(x)) f(x) lim f (x) c. If (Vxe S)g(x) # () and lim g(x)() then prove lim X-C XC 10 g(x) lim g(x) X-C
1. Let f:R → R be the function defined as: 32 0 if x is rational if x is irrational Prove that lim -70 f(x) = 0. Prove that limc f(x) does not exist for every real number c + 0. 2. Let f:R + R be a continuous function such that f(0) = 0 and f(2) = 0. Prove that there exists a real number c such that f(c+1) = f(c). 3 Let f. (a,b) R be a function...
Abstract Algebra Answer both parts please. Exercise 3.6.2 Let F be a field and let F = FU {o0) ( where oo is just a symbol). An F-linear fractional transformation is a function T: given by ar +b T(z) = cr + d ac). Prove that the set where ad-be 0 and T(oo) a/c, while T(-d/c) = o0 (recall that in a field, a/c of all linear fractional transformations M(F) is a subgroup of Sym(F). Further prove that if we...