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...
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 the ideal
I = < 6, 3 + 3 \sqrt{-17} >
in the ring Z [ \sqrt{-17} ].
Determine whether this ideal is prime or not.
Homework Equations
<18> = I^2
There is no element \alpha \in Z [ \sqrt{-17} ] such that 18 = \alpha^2
The Attempt at a Solution...
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)...
Homework Statement
Let p be an odd prime. Show that x^2 \equiv 2 (mod p) has a solution if and only if p \equiv 1 (mod 8) or
p \equiv -1 (mod 8)
The Attempt at a Solution
Ok, I figured the more of these I try, the better I'll get at them. Assuming that
x^2 \equiv 2 (mod p) has a...
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...
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...