Monthly Archives: December 2007

busy hectic manic crazy!

It’s currently 8:30 on the last friday of term, and I’ve been holed up in my room since just after lunch working on my dissertation. It has not been terribly fun, but it’s really my own fault for not spacing my workload out over the whole term properly.

Linda Marshall rang me earlier to let me know that I’d been accepted onto relay next year, and that I will be at Leicester Uni. That was pretty good news, it certainly takes a weight off my mind knowing exactly what’s going on next year. Now I just need to sort out my living arrangements and start raising the finances for the year (I’m sure you’ll be hearing more about that in the future!).

In a few minutes my house will be filled with CU people, which will be kinda nice. At that point I’m going to take a break from typing and pop downstairs to hang out, although not for too long as I really do need to get this work finished tonight.

After today it’s officially the holidays, which is great because it means I won’t have the pressure of this interim report deadline, and I’m going to be spending tomorrow hanging out with some guys from church, followed by hanging out with some other guys in the evening over a nice game or two of poker. I might even manage to squeeze in the mens breakfast at 7am, depending on how late I’m up working.

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Pure Evil

I am writing a dissertation currently. This is not a bad thing in itself, I actually rather enjoy writing, however there is a small problem with the subject, and general system for writing. I am a mathematician, and thus my dissertation contains a not insubstantial quantity of mathematics, furthermore, I am a pure mathematician, which means the maths is obscure and abstract, and is therefore represented in obscure and abstract ways.

As such, I have to make use of a typing system called LaTeX, which is extremely good in that it formats documents in a very standardised way, and is able to handle mathematical symbols very well, spacing them accurately and generally doing a good job of making things look correct. However the nature of the program is that instead of type a/b for a fraction, you have to type something like: $frac{a}{b}$, for it to show up correctly. Then you compile the stuff and it makes a nice pdf file which looks as it should.

This sounds like a good idea, and indeed it is. However the result of this approach to typing is that what I write looks like pure evil. Here is a short excerpt from what I have written so far in it’s raw pre-compilation format.

Let $i:K to K’$ be an isomorphism of fields, and let $Sigma$ be a splitting field for some $f$ over $K$, and $Sigma’$ be a splitting field for $i(f)$ over $K’$. Then there exists an isomorphism $j:Sigma to Sigma’$ such that $j|_k=i$. That is to say, the extensions $Sigma:K$ and $Sigma’:K’$ are isomorphic.

We have the following situation:$$xymatrix{
Kar[r]ar[d]_i & Sigmaar[d]^j \
K’ar[r] & Sigma’
and we must show first that $j$ exists, and then that it is in fact an isomorphism.

To begin with we will show by induction on the degree of $f$ that there exists a monomorphism $j:Sigma to Sigma’$ such that $j|_K = i$. $Sigma$ is a splitting field for $f$, so $f$ can be written as a polynomial over $Sigma$,
[ f(x) = (x – alpha_1) hdots (x – alpha_n), ]

with $alpha_1, hdots , alpha_n in Sigma$ being the zeros of $f$ in $Sigma$. There will then be a minimum polynomial, $m_{alpha_{1}}$ of $alpha_1$, and this $m_{alpha_{1}}$ will be and irreducible factor of $f$. This $i(m_{alpha_{1}})$ will also divide $i(f)$, and hence will split over $Sigma’$, so it can be written;
[ i(m_{alpha_{1}}) = (x – beta_{alpha_{1}1}) hdots (x – beta_{alpha_{1}r}), ]

with $beta_{alpha_{1}1}, hdots , beta_{alpha_{1}r} in Sigma’$. We also know that $i(m_{alpha_{1}})$ is irreducible over $K’$, so it is a minimum polynomial of $beta_{alpha_{1}1}$ over $K’$. Hence by ref{th3.9}, we have an isomorphism
[ j_1 : K(alpha_1) to K'(beta_{alpha_{1}1}) ]

such that $j_1|_K = i$ and $j_1(alpha_1) = beta_{alpha_{1}1}$. It is clear that $Sigma$ is also a splitting field of $g = f/(x-alpha_1)$ over $K(alpha_1)$, and so by induction we can form an isomorphism
[ j:K(alpha_1, hdots, alpha_n) to K'(beta_{alpha_{1}1}, hdots , beta_{alpha_{n}1}). ]

As we saw in Definition 1.3, $K(alpha_1, hdots, alpha_n)$ is precisely the splitting field for $f$ over $K$, $Sigma$. And since each $beta_{alpha_{1}1}, hdots, beta_{alpha_{n}1}$ is a zero of $i(f)$, $K'(beta_{alpha_{1}1}, hdots, beta_{alpha_{n}1}) subseteq Sigma’$. Hence $j:Sigma to Sigma’$ is a monomorphism, and by construction $j|_{K(alpha_{1})} = j_1$. So we now have that $j|_K = i$.

All that remains to be shown is that this $j$ is in fact an isomorphism. This follows easily, since $j(Sigma)$ is a splitting field for $i(f)$ over $K’$, and is contained in $Sigma’$, and $Sigma’$ is also a splitting field for $i(f)$ over $K’$. So we have that $j(Sigma) = Sigma’$, so $j$ is surjective, and hence is an isomorphism.

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