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I have edited the first post....i left out p is prime and we want to see show that the congruence is not solvable.

Homework Statement Show x^2 + (p+1)/4 \equiv 0 (\mod p) where p \equiv 3 (\mod 4) and p is prime is not solvable. Homework Equations Legendre's and Jacobi symbol, congruences The Attempt at a Solution Noticing that x^2 \equiv -(k+1) (\mod p) when p = 4k + 3 ? Now (-1/p)(k+1/p) should tell...
3. ### Application of cyclotomic polynomial

Homework Statement Show that {p \choose k} = \sum^{k+1}_{i=1} {p-i \choose p-k-1} where \forall k < p \in \mathbb{Z} and p a prime. Homework Equations This is part (b) to a problem. Part (a) is showing that 1 + x + x^2 + \cdots + x^{p-1} is irreducible in \mathbb{Q}[x]. The Attempt...
4. ### R[x] UFD, then R UFD

Homework Statement Assuming R is an integral domain. If the polynomial ring of one variable, R[x], is a unique factorization domain, then R is a unique factorization domain. The Attempt at a Solution Should be straightforward...so much so that I don't know how to start...probably with...
5. ### Frattini subgroup

Homework Statement G is a finite p-group, show that G/ \Phi (G) is elementary abelian p-group. Homework Equations \Phi (G) is the intersection of all maximal subgroups of G. The Attempt at a Solution By sylow's theorem's we have 1 Sylow p-subgroup which is normal, call P. Then the order...
6. ### Normalizer acts on stabilizer

Homework Statement G acts transitively on S and let H be the stabilizer of s. Show that the normalizer of H, call it N, acts transitively on the fixpoints of H, call it F, where s is some element in S. Homework Equations Two different ways of showing this: Either we show the orbit for any...