"Along the way, [Galois] introduced some amazingly original and powerful concepts, which form the framework of much algebraic thinking to this day. Although Galois did not work explicitly in axiomatic algebra (which was unknown in his day), the abstract notion of algebraic structure is clearly prefigured in his work." (Pinter)
I mean I'd consider being more applications of number theory to compsci but I've had an attempted run with physics and it seriously didn't blow over well
@0celo7 I mean if we're talking about being useful as an end in itself versus usefulness with the end of having a good career, I don't think I'm particularly interested in the former
@LeakyNun ok. call it G. there's a map from G to S_2, permutations of {i,-i}. It's surjective because of conjugation. Now we must find the kernel K. Then K permutes the four roots of 2, the only question is how.
Like, in principle I'd rather spend time doing what I want to do. If I am gonna sacrifice that, it won't be for abstract usefulness so much as being broke. So finance would be an open option, though admittedly I'd rather try to see if something like data science or being an actuary is an option ahead of like, Wall Street style stuff. The latter just seems to be a bit... dodgy
@LeakyNun Say $\alpha$ is an automorphism that fixes $i$ and sends $\sqrt[4]{2}$ to (wlog) $i\sqrt[4]{2}$. If it exists, it is fully determined, has order four, so must generate the kernel $K$.
a relevant point: "In 1847, Gabriel Lamé outlined a proof of Fermat's Last Theorem based on factoring the equation $x^p + y^p = z^p$ in complex numbers, specifically the cyclotomic field based on the roots of the number 1. His proof failed, however, because it assumed incorrectly that such complex numbers can be factored uniquely into primes, similar to integers. This gap was pointed out immediately by Joseph Liouville, who later read a paper that demonstrated this failure of unique factorisation, written by Ernst Kummer.
the following remark shows up on Wikipedia: "For a square-free positive integer n, the quadratic integer ring Z[sqrt(-n)] will fail to be a UFD unless n is a Heegner number." (these are 1, 2, 3, 7, 11, 19, 43, 67, 163. )
@anon hmm, my pathway is a little bit different: the degree of the polynomial is $4$, so the group must be a subgroup of $S_4$, which means it is $D_8$
@0celo7 if I reorganize it to $$(\xi\cdot X)h(\dot v_1,\dot v_2)=m(\xi,X)(\dot v_1,\dot v_2)+h(\xi\otimes \dot v_1\cdot X,\dot v_2)+h(\dot v_1,\xi\otimes \dot v_2\cdot X)$$ it sorta reminds me of the Leibniz rule
Let's call $\sigma$ a generator of the Galois group. Since $\sigma^2$ is the unique element of order $2$, and complex conjugation has order $2$, we find that $\sigma^2$ must be complex conjugation. So $\sigma^2(i)=-i$. There are two possibilities for $\sigma(i)$, it's either $i$ or $-i$, but both cases lead to $\sigma^2(i)=i$, a contradiction.
[Random] As I see more and more textbooks and attempted explanations of math, the more I realize just how good the MIT open courseware materials I studied from really are:
This material breaks the "Wikipedia barrier" between an interested high school student or high school graduate and someone who can actually read math and understand what is meant.
@Semiclassical Interesting Remark: Nother's theorem assumes the Lagrangian is first order in derivatives, but the GR Lagrangian isn't. Moreover, the GR Lagrangian is essentially unique so this is a fundamental issue.
@Semiclassical Alright. I guess the motivation should take place on a curved $\Bbb R^4$, and then in the end you pull a vast generalization and define the stuff like that in the general setting
And the integrals of these things turn out to be well-defined
my book gives a description for an algo that finds the MST of a graph: "Grow a tree one edge at a time by adding the minimum weight edge possible to the tree, making sure that you have a tree at each step." to my understanding, this is known as Prim's algorithm?
