Hello HN! Thank you for upvoting this post to the front page. I'm quite delighted to see this post about a relatively obscure lemma from Galois theory make it to the front page.
This post is primarily intended as notes on the subject. The theorems and proofs involved in Galois theory can often feel too abstract, so I find it helpful to work through them with concrete examples. This is one such example that I wanted to archive for future reference on my website. If you spot any errors, please let me know.
Thank you for sharing your notes—everyone needs a different push/viewpoint to make key concepts click, and, the more expositions are out there, the more likely someone will be to find the one that's right for them!
Is M^* Stewart's notation? I'm much more used to seeing it used for a dual space than a group of automorphisms. Conventions aside, it's a bit unfortunate in that it doesn't specify the field on which we're acting! I think that the notation Aut(L/M), or Gal(L/M) for a Galois extension, is more common.
Thank you! Yes, M^* is Stewart's notation. Indeed this notation is "lossy" in the sense that it loses information about what the extension field is. The extension field is usually introduced once for every example or theorem and then M^* is used as a shorthand notation. For example, in my post, the extension field is introduced like this:
"The notation M^* denotes the group of all M-automorphisms of L with composition as the group operation."
By the way, Stewart uses the notation Γ(L/M) too at several places but in this specific lemma, the notation M^* is used and therefore my post too uses the same notation.
Thanks! Yes, I handcraft all my HTML and CSS. I'm glad you noticed the HTML and liked it. I find great joy in crafting my website by hand. It's like digital gardening. I grow all my HTML and CSS myself. It's all 100% organic and locally sourced!
I rarely ^U nowadays, but your site was so clean that I couldn't resist!
Just as a side note : when writing html5 by hand, you can use the full power of the language, most notably optional tags (no need to write html, body, etc) and auto-closing tags (no need to close p, li, td, etc). You may get something even crispier!
> Notice that, when writing html5 by hand, you can use the full power of the language, most notably optional tags (no need to write html, body, etc) and auto-closing tags (no need to close p, li, td, etc). You may get something even crispier!
Yes! In fact, sometime back I wrote a little demo page to show the minimal (but not code-golfed) HTML we can write such that it passes validation both with the Nu HTML Checker and HTML Tidy.
That said, when writing my own posts, I prefer keeping optional and closing tags intact. Since I use Emacs, I can insert and indent closing tags effortlessly with C-c /. It's a bit like how some people write:
10 PRINT"HELLO
But I've always preferred:
10 PRINT "HELLO"
I find the extra structure more aesthetically pleasing.
This post is primarily intended as notes on the subject. The theorems and proofs involved in Galois theory can often feel too abstract, so I find it helpful to work through them with concrete examples. This is one such example that I wanted to archive for future reference on my website. If you spot any errors, please let me know.