Fields, Polynomials, GCD and Divisibility - 1A) Let F be a field and let f(x), g(x) and h(x) be polynomials in Fx. Prove that gcd(f(x),g(x)) = 1F and f(x) divides g(x)h(x), then f(x) divides h(x).
1B) Let F be a field and l ...
Divisibility of Sequences - Let m, n be in N, with m, n >= 1 and n odd.
Let S_m = 1^n + 2^n + 3^n + ... + m^n.
Prove that S_m is divisible by 1+2+...+m.
Ring proof - Let R be a commutative ring with 1. Prove that if (a,b)=1 and a divides bc, then a divides c. More generally, show that if a divides bc with nonzero a,b then a/(a,b) divides c. (Here (a,b) denotes the ...
Minimal Polynomial - Hello, I need a detailed solution to the attached problem:
Let A be a finite dimensional F-algebra, a is in A a unit of finite order m.....
Hint for part (a):
p | x^k - 1 <=> x^k = 1 in F[x] ...
Proofs by induction - Show by induction that:
a) Sum(n^2/[(2n-1)*(2n+1)],n=1..) = n(n+1)/[2*(2n+1)]
b) Sum( r/(r-1)!, r=1..n) = 1-1/(n+1)!
c) n^3+3n^2-10n is divisible by 3
d) 4^(2n+1) + 3^(n+2) is divi ...