Recent content by Zoe-b

Hmmn I think I was getting confused with the logic of what I was trying to do then. I can take f(at) = a |t| (where |t| is its norm). Then if every functional vanishes at t then the extension of f, g satisfies g(t) = 0 = |t| and by positive definiteness t is zero.

Possibly in the right direction, but unfortunately not at speed :P I want to find a linear functional f defined on M s.t. f vanishes at t (so that its Hahn-Banach extension will satisfy the given property).. but if f vanishes at t then it vanishes on the whole of M, which is not particularly...

Homework Statement To clarify- this isn't a homework problem; its something that's stated as a corollary in my notes (as in the proof is supposed to be obvious) and I haven't yet managed to prove it- I'm probably just missing something! Would appreciate a hint or a link to where I might find...

Ok thank you I thought that was the case but then got confused by questions where the splitting field seemed to be the same for different examples :)

Fantastic- I have a more general question which as yet I've been unable to find the answer to in a textbook.. Does the galois group of a polynomial depend purely on its splitting field? Or is it in some way connected to the polynomial itself? For example, if two polynomials have different roots...

Think I've got it now- the intermediate field is Q(sqrt(5)) which is fixed by the subgroup {e,t} where e is the identity and t sends w to w^4, w^2 to w^3, that is, t is equivalent to complex conjugation. Thank you!

Ok... true. Will an intermediate field be one over which x^4 + x^3 + x^2 + x + 1 splits into two quadratics? or is the polynomial irrelevant for this...?

Homework Statement I'm trying to find the galois group of x^5 - 1 over Q, and then for each subgroup of the galois group identify which subfield is fixed. Homework Equations The Attempt at a Solution If w = exp(2*I*PI/5), then the roots not in Q are w, w^2, w^3, w^4. Its fairly...

Sorry for the bump but I'm still stuck on this- can anyone help? Thanks Zoe

Homework Statement The question says: Find the degrees of the splitting extensions of the following polynomials, and show that in each case the number of automorphisms of the splitting field is at most the degree of the extension. i) x^3 - 1 over Q (3 others) Homework Equations...

Homework Statement So I have a question that says: Let T:R -> S be a ring homomorphism, show that if J is a prime ideal of S, then T-1(J) := { r in R s.t. T(r) is in J) is a prime ideal of R. (I've done this bit) It then says: Give an example where J is maximal but T-1(J) is not...

Yeah done it now, thanks! I guess I did use pythagoras- Just conceptually not that happy with drawing triangles when the elements I'm using aren't necessarily vectors? Anyway, done it using just inner product notation and now happy :)

Thanks for the replies- Hilbert spaces aren't on my course yet so I don't think that's the way to go. I'm trying to use the second hint and do it with a diagram but not getting that far- also this is in any real inner product space not necessarily R^n..

I haven't tried it but it depends whether the eigenvalue two has two linearly independent eigenvalues or not- getting a computer to find one is not going to help here. You need to write out the set of simultaneous equations A(v) = 2v and see if when solving them you get two linearly...

Homework Statement Let U be the orthogonal complement of a subspace W of a real inner product space V. Have already shown that T is a projection along a subspace W onto U, and that V is the direct sum of W and U. The questions now says: show ||T(v)|| = inf (w in W) || v - w ||...