Recent content by ElDavidas

I just graduated recently with a 2:1 in Mathematics and am thinking of studying a Masters course in Medical Physics in 2008. I got a welcoming response from Aberdeen University (http://www.biomed.abdn.ac.uk/Courses/medphys.html) and am thinking of studying there. However, I'm trying to think...

The roots of that polynomial are the primative roots \zeta and \zeta^5. How would I now use this information?

Homework Statement Let \zeta be a primative 6-th root of unity. Set \omega = \zeta i where i^2 = 1. Find a square-free integer m such that Q [\sqrt{m}] = Q[ \zeta ] Homework Equations The minimal polynomial of \zeta is x^2 - x + 1 The Attempt at a Solution I was...

Homework Statement Take L = \left(\begin{array}{ccc}0 & -a & -b \\b & c & 0 \\a & 0 & -c\end{array}\right) where a,b,c are complex numbers. Homework Equations I find that a basis for the above Lie Algebra is e_1 = \left(\begin{array}{ccc}0 & -1 & 0 \\0 & 0 & 0 \\1 & 0 &...

Take L to be a subspace of sl (2,R). (R is the real numbers) L = \left( \begin{array}{ccc} 0 & -c & b\\ c & 0 & -a\\ -b & a & 0 \end{array}\right) The basis elements of L are e_1 = \left( \begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & -1\\ 0 & 1 & 0 \end{array}\right)...

Ah, I see now! Thanks.

I'm not sure. Could you use the Euclidean Algorithm? I've also been trying to find a contradiction by assuming that gcd (x,p) \neq 1 .

ok, so far my solution is as follows: x^3 \equiv a (mod p) Then x \equiv a^{\frac{1}{3}} (mod p) So by Fermat, x^{p-1} \equiv 1 (mod p) or a^{\frac{1(p-1)}{3}} \equiv 1 (mod p) Would I need to say gcd (x,p) = 1? If so, how is this true?

Homework Statement Let p be a prime number such that p \equiv 1 (mod 3) Let a be an integer not divisible by p. Show that if the congruence x^3 \equiv a (mod p) has a solution then a^{\frac{p - 1} {3}} \equiv 1 (mod p) The Attempt at a Solution Right, I'm not sure how...

Sorry, what do you mean by this?

I've been looking at this example for a while now. Could someone help? "Take the functional to be J(Y) = \int_{a}^{b} \( \alpha Y'^2 + \beta Y^2) dx For this F(x,y,y') = \alpha y'^2 + \beta y^2 and p = \frac{ \partial F}{\partial y'} = 2 \alpha y' \Rightarrow y' =...

Sorry, I don't understand this. Is finding the fixed point of \sigma (\alpha) something to do with the sign in between the 2 and \sqrt{2} not changing?

Homework Statement i) Find the order and structure of the Galois Group K:Q where K = Q(\alpha) and \alpha = \sqrt{2 + \sqrt{2}}. ii)Then for each subgroup of Gal (K:Q), find the corresponding subfield through the Galois correspondence. Homework Equations I get the minimal...

Homework Statement Show Q(\sqrt{p},\sqrt{q}) = Q(\sqrt{p} + \sqrt{q}) Homework Equations p and q are two different prime numbers The Attempt at a Solution I can show \sqrt{p} + \sqrt{q} \in Q(\sqrt{p},\sqrt{p}) I have trouble with the other direction though, i.e...

Of course, that would be helpful. g(x)(x-1) = (x^9-1) and the roots are \alpha existing in the complex numbers such that \alpha^9 = 1