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:

    Thanks for the linkage.

  11. Axel says:

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

  12. Alon Levy says:

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

  13. 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.

  14. Phil says:

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

  15. 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. ;.;

  16. Cairnarvon says:

    (And apparently backslashes are stripped in your comments, so that link leads to a broken-looking formula. Try this, then.)

  17. […] just discovered via Abstract Nonsense that there’s now built-in support for LaTeX in WordPress! Now that’s way […]

  18. acomplia says:

    Strange text in alt tag

  19. 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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