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...
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...
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...
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...
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...