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.

This entry was posted on Saturday, February 17th, 2007 at 9:26 pm and is filed under Mathematics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

20 Responses to “WordPress Installs LaTeX (and I Classify Finite Fields)”

  1. Tyler DiPietro Says:
    February 17, 2007 at 10:08 pm | Reply

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

    Thanks for nooze, Alon.

  2. Richard Says:
    February 17, 2007 at 10:20 pm | Reply

    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:
    February 17, 2007 at 10:21 pm | Reply

    Apparently not.

  4. rod. Says:
    February 17, 2007 at 11:05 pm | Reply

    Oh yes! It works on comments as well :-)

    Just check this out:

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

  5. rod. Says:
    February 17, 2007 at 11:06 pm | Reply

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

  6. D. Eppstein Says:
    February 18, 2007 at 12:29 am | Reply

    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:
    February 18, 2007 at 12:44 am | Reply

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

  8. Phil Says:
    February 18, 2007 at 6:04 am | Reply

    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:
    February 18, 2007 at 6:06 am | Reply

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

  10. whig Says:
    February 18, 2007 at 10:32 am | Reply

    Thanks for the linkage.

  11. Axel Says:
    February 18, 2007 at 5:41 pm | Reply

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

  12. Axel Says:
    February 18, 2007 at 5:44 pm | Reply

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

  13. Alon Levy Says:
    February 18, 2007 at 7:35 pm | Reply

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

  14. Alon Levy Says:
    February 18, 2007 at 7:37 pm | Reply

    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:
    February 19, 2007 at 7:08 am | Reply

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

  16. Cairnarvon Says:
    February 20, 2007 at 11:31 am | Reply

    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:
    February 20, 2007 at 11:33 am | Reply

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

  18. I am … unhindered by talent » Blog Archive » LaTeX for WordPress! Says:
    February 21, 2007 at 1:22 pm | Reply

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

  19. acomplia Says:
    February 16, 2008 at 3:10 pm | Reply

    Strange text in alt tag

  20. /zyloprim/ Says:
    February 25, 2008 at 8:15 pm | Reply

    yup, i see it too

Leave a Reply