Project #18158 - Galois Theory

4 Questions

1) Show that x^6+3 is irreducible over Q, but is not irreducible over Q(w) where w is a primitive sixth root of unity.

2) Suppose that q is a prime, Char K=/=q and that x^q - pheta is irreducible in K[x]. Let w be a primitive qth root of unity, and let [K(w):K]=j. Show that the Galois group of x^q-pheta can be generated by elements sigma and tau satisfying: sigma^q=tau^j=1, sigma^k*tau=tau*sigma, where k is a generator of the mulitplicative group Zq*.

3) Suppose that L:K is a Galois extension of degree n with Galois group G. If x is in L, let tr(x)=sum sigma(x) where sigma is in G, N(x) = product sigma(x) where sigma is in G. The mapping M is the norm. Suppose that b is in L, has a minimal polynomial x^r-a1*x^(r-1)+...+(-1)^r*ar. Show that tr(b)=(n/r)*a1 and N(b)=ar^(n/r)

4) Suppose that L:K is a Galois extension of degree n with cyclic Galois group generated by tau

(i) Suppse that a=b/tau(b). Show that N(a)=1

(ii) Suppose that N(a)=1. Let c0=a1, c1=a*tau(c0), c2=a*tau(c1),..., cn-1=a*tau(cn-1). Show that there exists u such that b=c0*u+c1*tau(u)+...+cn-1*tau^(n-1)(u)=/=0. Show that a=b/tau(b)

Subject Mathematics
Due By (Pacific Time) 12/02/2013 12:00 am
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