## WordPress Installs LaTeX (and I Classify Finite Fields)

Hat-tip to Rod: WordPress has a new $\LaTeX$ feature for mathematicians like me. I like Whig’s take on it the most; mine is going to be more prosaic.

Let $K = \mathbb{F}_{p} = \mathbb{Z}/p\mathbb{Z}$, and let $L$ be an extension of $K$ of degree $n < \infty$. Then $L$ is unique among all degree-$n$ extensions. To see why, note that $|L| = p^{n} \Rightarrow \forall a \in L^{*}: a^{p^{n}-1} = 1$, so $L$ is the splitting field for $x^{p^{n}-1} - 1$ over $K$. As splitting fields are unique, the result follows.

Conversely, for every $n$, there exists a field $L$ with $[L:K] = n$. To prove that, it’s enough to prove that the splitting field of $x^{p^{n}-1} - 1$ has exactly $p^{n}$ elements. The polynomial has a nonzero constant, so it doesn’t have zero as a root.

If the field has fewer than $p^{n}-1$ nonzero elements, then it must have repeated roots. But its derivative is $(p^{n}-1)x^{p^{n}-2} = -x^{p^{n}-2}$, whose sole irreducible factor, $x$, doesn’t divide $x^{p^{n}-1} - 1$, contradicting the result that the polynomial has a repeated root.

To see that it doesn’t have more than $p^{n}$ elements, note that the roots of the polynomial together with 0 form a field. This is because if $a^{p^{n}} = a$ and $b^{p^{n}} = b$, then $(ab)^{p^{n}} = ab$ clearly, and $(a + b)^{p^{n}} = a + b$ by repeatedly applying the Frobenius automorphism.

That completes the classification of finite fields, which states that there is a unique finite field for each prime power order, and no finite field with an order that isn’t a prime power.

On another note, be nice to me. If you ask politely, I might go back to my earlier math posts and edit them to incorporate $\LaTeX$.

### 24 Responses to WordPress Installs LaTeX (and I Classify Finite Fields)

1. YAY! No more ad hoc notational jury-rigging for me!!!!!

Thanks for nooze, Alon.

2. Richard says:

Hmmm, does this work in comments as well?

$\widetilde{\phi}_{_\scr{X}} \cap \cal{M} \;\supsetneqq\; \widetilde{\phi}_{_\scr{Y}} \cap \cal{M}$

3. Richard says:

Apparently not.

4. rod. says:

Oh yes! It works on comments as well 🙂

Just check this out:

$\widetilde{\phi}_{_\scr{Y}} \cap \cal{M}$

5. rod. says:

Ok, it didn’t work… but this should work: $\LaTeX$

6. D. Eppstein says:

Yay! Now the mathematical typography on blogs will be as bitmapped and as mismatched in size and font to the body text as it is on Wikipedia!

7. rod. says:

hey, better a size-mismatched math type than nothing, lol 😉

8. Phil says:

In a previous life I used LaTeX to write math, but now that we are in the XML-based 21st century it has become, like the PDP-11, a relic. Or should be anyway.

Why not turn entire posts/articles into single s and be done with it?

9. Phil says:

(that is) Why not turn entire posts/articles into single <img src=”….bmp>s and be done with it?

10. whig says:

11. Axel says:

Let’s try…
$P(X=k)&=e^{-\lambda}\: \frac{\lambda^k}{k!} \quad \text{for$k=0,1,\ldots$}$

12. Axel says:

Hmm,
$P(X=k)&=e^{-\lambda}\: \frac{\lambda^k}{k!}$

13. Alon Levy says:

I’m not sure why the formula didn’t parse.

14. Alon Levy says:

Phil, in BMP format even a short post like this one will weigh over 500 KB. My longer math posts will go into the megabytes.

15. Phil says:

My point is that LaTeX (at least it was “markup”) is ancient history, like the Atari 🙂

16. Cairnarvon says:

Bah. I’m stuck using Mimetex (which works, but is ugly as hell), because I can’t install software on a shared server. Unfair. ;.;

17. Cairnarvon says:

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19. acomplia says:

Strange text in alt tag

20. /zyloprim/ says:

yup, i see it too

21. You actually make it seem so easy with your presentation but I find this matter to be really something that I think I would never understand. It seems too complex and extremely broad for me. I’m looking forward for your next post, I’ll try to get the hang of it!

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