tag:blogger.com,1999:blog-6837159629100463303.post5524643365182245222..comments2021-07-22T00:29:53.205-07:00Comments on Information Transfer Economics: Leeches, a rantJason Smithhttp://www.blogger.com/profile/12680061127040420047noreply@blogger.comBlogger16125tag:blogger.com,1999:blog-6837159629100463303.post-5428377283282172242016-01-21T12:51:26.446-08:002016-01-21T12:51:26.446-08:00Ha!... well, I encountered them again this month ...Ha!... well, I encountered them again this month when asked to size a <a href="https://en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction" rel="nofollow">Reed-Solomon encoder/decoder</a> in terms of computational requirements (I looked at FPGAs). I've seen Galois fields discussed in processor instruction set documentation. <a href="http://www.ti.com/lit/ug/sprufe8b/sprufe8b.pdf" rel="nofollow">Here's an example.</a> I wonder why they latched onto that name?Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-46935656042576757562016-01-21T12:04:55.691-08:002016-01-21T12:04:55.691-08:00It just seems like this would lead to confusion.
...It just seems like this would lead to confusion.<br /><br />Wikipedia just re-directs Galois field to finite field:<br /><br /><a href="https://en.wikipedia.org/w/index.php?title=Galois_field&redirect=no" rel="nofollow">https://en.wikipedia.org/w/index.php?title=Galois_field&redirect=no</a>Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-52738312696002986292016-01-21T12:00:54.191-08:002016-01-21T12:00:54.191-08:00Thanks. I'd never heard of Sylow theorems befo...Thanks. I'd never heard of Sylow theorems before.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-67733548920624463962016-01-21T11:58:49.913-08:002016-01-21T11:58:49.913-08:00To clarify:
Galois theory has to do with the auto...To clarify:<br /><br />Galois theory has to do with the automorphism groups of field extensions (Galois groups), whereas Galois fields are just another name for finite fields -- one that I don't remember using, but maybe we did -- which you can talk about independently of Galois theory.Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-91088870707356216562016-01-21T11:43:05.717-08:002016-01-21T11:43:05.717-08:00The relationship appears to be that they call fini...The relationship appears to be that they call finite fields Galois fields, but the results in error correction have more to do with Sylow theorems ... but maybe there is more to it than that ...<br /><br /><a href="https://en.wikipedia.org/wiki/Sylow_theorems" rel="nofollow">https://en.wikipedia.org/wiki/Sylow_theorems</a>Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-25187582121277592422016-01-21T10:24:48.784-08:002016-01-21T10:24:48.784-08:00Entertaining rant Jason. I've encountered a Ga...Entertaining rant Jason. I've encountered a Galois field in error correction (and I'm familiar w/ Galois' brief life story, and his role with proving no guarantee of closed form solution to quintics and above), but after reading the comments here I can see that Galois fields probably have little or nothing to do with Galois theory. I certainly have no idea! It's always fun to encounter vast new regions of my ignorance.Tom Brownhttps://www.blogger.com/profile/17654184190478330946noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-91116027164714617572015-05-18T18:13:00.014-07:002015-05-18T18:13:00.014-07:00I think we have different definitions of "Gal...I think we have different definitions of "Galois theory" -- your definition seems to be the mathematics used by Evariste Galois while my definition is the mathematics of Galois groups.<br /><br />Galois theory ≠ group theory<br /><br />Also Ruffini and Lagrange came up with the ideas about permutation groups that Galois extended. I guess he came up with the name "group", but if you've ever had a class on group theory you know that there are lots of theorems due to Lagrange.Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-59282158210464813912015-05-18T17:36:32.616-07:002015-05-18T17:36:32.616-07:00Aaargh. Frankly, I find such anti-historicism horr...Aaargh. Frankly, I find such anti-historicism horrifying. It is like criticizing Aristotle & Shakespeare for being full of platitudes and cliches, or worse not noticing that your mom bought the cliches you are speaking in from A & S. Galois Theory is the first and most easily graspable instance of enormously important concepts in mathematics and physics - giving birth to group theory, field theory (not applying one to another) and abstract algebra and in turn being founded on high-school algebra as a treatment by Arnol'd or Abhyankar might give. Weyl was talking about basically all the mathematics of the rest of the 19th century and the Galoisian spirit of the 20th (Hilbert/Noether/ Van der Waerden/Bourbaki). And both in the abstract as above and the concrete, the conscious way that Galois's theory inspired Riemann Surfaces/Complex Analysis/Function fields and through that the development of the fundamental group by Poincare on the one hand, (people more knowledgeable than me have said that some of this seems to be in Galois already) and the continuous world of Lie theory to which Weyl made so many contributions. Weil, Grothendieck etc of course made it clear that the algebraic topological fundamental group was not merely analogous to Galois theory, e.g. of function fields/ riemann surfaces which it had descended from, but unified, subsumed both in general theories. Klein's solution of the quintic, a quintic formula, in his classic The Icosahedron IIRC was another return to the roots / further development, as was all of his & Poincare's work in modular forms / discrete subgroups of Lie Groups to be anachronistic. Elliptic curve cryptography, the elliptic curves stemming from the RS / galois theory stream of Galois, over the Galois (Finite) Fields stemming from another, are used in everyone's cell phones.<br /><br /><i>It has no application in physics or any other science.</i> No, everything is an application of it. Weyl's gauge theory, sheaf / fibre bundle theory (generalized from the discrete, galois case) is inconceivable without it.Calgacushttps://www.blogger.com/profile/06031818010224747000noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-65541540815659178842015-05-18T13:37:34.496-07:002015-05-18T13:37:34.496-07:00Weyl thinks Galois theory is a good example of mat...Weyl thinks Galois theory is a good example of math, not an end in and of itself:<br /><br /><i><br />Turning from physics to mathematics, [Weyl] gives an extraordinarily concise epitome of Galois theory, leading up to the statement of his guiding principle: "Whenever you have to do with a structure-endowed entity, try to determine its group of automorphisms".<br /></i><br /><br /><a href="http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Weyl.html" rel="nofollow">http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Weyl.html</a>Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-22633115458296978732015-05-18T13:26:16.521-07:002015-05-18T13:26:16.521-07:00Galois life was pretty epic, and his idea to look ...Galois life was pretty epic, and his idea to look at the connections between field and group theory were a good example of great mathematical works. But contra Weyl, the results themselves have little application and many mathematicians go through undergraduate (and even graduate) education without ever studying it. It has no application in physics or any other science.<br /><br />I studied it as part of my math degree. It's a great example of <b>doing mathematics</b>, but the results themselves have no real world application -- exactly the point I was trying to make about economics! It's some really nice theory with some fun math, but no (apparent) real world application!<br /><br />What specifically is Weyl basing his claims on?<br /><br />Here's a modern math professor asking whether Galois theory is necessary:<br /><br /><a href="http://mathoverflow.net/questions/34125/is-galois-theory-necessary-in-a-basic-graduate-algebra-course" rel="nofollow">http://mathoverflow.net/questions/34125/is-galois-theory-necessary-in-a-basic-graduate-algebra-course</a><br /><br />Galois is super cool, though!Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-48470402676146952962015-05-18T13:09:46.313-07:002015-05-18T13:09:46.313-07:00It feels like studying Galois theory -- a topic fr...<i>It feels like studying Galois theory -- a topic from mathematics that seems to just exist to solve a single problem (why is there no quintic equation) for which there is no real-world application.</i><br /><br />Hard to think of a worse example than Galois Theory.<br />Hermann Weyl, <i>Symmetry</i>, ρ 138.<br />"Galois' ideas, which for several decades remained a book with seven seals but later exerted a more and more profound influence upon the whole development of mathematics, are contained in a farewell letter written to a friend on the eve of his death, which he met in a silly duel at the age of twenty-one. This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind."Calgacushttps://www.blogger.com/profile/06031818010224747000noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-8578988841729627522015-05-13T15:43:19.288-07:002015-05-13T15:43:19.288-07:00Except in the case where they are caused by change...Except in the case where they are caused by changes in productivity (i.e. RBC) ...<br /><br />I think the constructibility proofs came much later ... when people were trying desperately to figure out other uses for it :)Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-85652724828084721482015-05-13T14:34:50.361-07:002015-05-13T14:34:50.361-07:00Rcessions are caused by a demand or supply shock. ...Rcessions are caused by a demand or supply shock. Everyone knows that!<br /><br />But don't ask me what a demand/supply shock is because I don't know.<br /><br />You can use Galois Theory to prove that squaring the circle is impossible, but I admit that this is not a very real-worldish application.Mnoreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-78989371038425448802015-05-13T11:53:31.474-07:002015-05-13T11:53:31.474-07:00Ha! Thanks Todd.Ha! Thanks Todd.Jason Smithhttps://www.blogger.com/profile/12680061127040420047noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-56924681807697875332015-05-12T21:49:25.118-07:002015-05-12T21:49:25.118-07:00Preach it, brother! Could not agree more.Preach it, brother! Could not agree more.Todd Zorickhttps://www.blogger.com/profile/10976192775890569092noreply@blogger.comtag:blogger.com,1999:blog-6837159629100463303.post-33090096359732980102015-05-12T21:49:19.372-07:002015-05-12T21:49:19.372-07:00Preach it, brother! Could not agree more.Preach it, brother! Could not agree more.Todd Zorickhttps://www.blogger.com/profile/10976192775890569092noreply@blogger.com