Timeline for Ideas for introducing Galois theory to advanced high school students
Current License: CC BY-SA 4.0
22 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 21, 2021 at 0:26 | answer | added | Tony Gonzalez | timeline score: 2 | |
| Dec 18, 2021 at 6:31 | answer | added | Mozibur Ullah | timeline score: 4 | |
| Dec 17, 2021 at 20:37 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Dec 17, 2021 at 15:40 | answer | added | Kapil | timeline score: 2 | |
| Dec 17, 2021 at 10:50 | answer | added | Ben McKay | timeline score: 5 | |
| Dec 17, 2021 at 9:36 | vote | accept | user929304 | ||
| Dec 16, 2021 at 22:03 | comment | added | Abdelmalek Abdesselam | For more details, there is also the book by Eugen Netto "The Theory of Substitutions and Its Applications to Algebra" full text available from Google Books. | |
| Dec 16, 2021 at 21:56 | comment | added | Abdelmalek Abdesselam | I recommend arxiv.org/abs/1301.7116 | |
| Dec 16, 2021 at 20:41 | answer | added | Max Lonysa Muller | timeline score: 5 | |
| Dec 16, 2021 at 19:41 | answer | added | R.. GitHub STOP HELPING ICE | timeline score: 11 | |
| Dec 15, 2021 at 21:11 | history | became hot network question | |||
| Dec 15, 2021 at 17:27 | comment | added | Louis D | I would recommend looking at the book "Galois Theory for Beginners: A Historical Perspective" by Jorg Bewersdorff. Even if it doesn't suit your needs, it may give you some ideas. | |
| Dec 15, 2021 at 16:55 | answer | added | Alexandre Eremenko | timeline score: 23 | |
| Dec 15, 2021 at 16:48 | comment | added | Timothy Chow | Galois Theory by Harold Edwards, despite being in the Graduate Texts in Mathematics series, is worth looking at, because Edwards carefully studied Galois's original memoir, and said, "I saw that the modern treatments of Galois theory lacked much of the simplicity and clarity of the original." I'm not suggesting that your students read Edwards directly, but you may find some useful ideas in his book that aren't easy to find elsewhere. | |
| Dec 15, 2021 at 15:55 | comment | added | R.P. | Another issue that deserves more attention imo is the casus irreducibilis among third-degree equations. These are cubic equations whose real solutions cannot be expressed in terms of "real radicals", in other words the root itself is real, but in order to express it in terms of radicals you need the complex numbers. In a way, I feel this fact seriously qualifies the truism that third degree equations can be solved in terms of radicals. Proving it is a nontrivial exercise in field theory, maybe it is worth giving a sketch; a proof can be found in Van der Waerden's Algebra. | |
| Dec 15, 2021 at 15:44 | answer | added | David E Speyer | timeline score: 70 | |
| Dec 15, 2021 at 14:34 | comment | added | R.P. | In the same vein, I remember that I read Pollard & Diamond (see amazon.com/Theory-Algebraic-Numbers-Dover-Mathematics/dp/…) as a high-school student. It steers clear from full-blown Galois theory, but it does discuss a lot of the theory leading up to and foreshadowing it (including many examples of the sort Mr. Strickland is alluding to). | |
| Dec 15, 2021 at 14:25 | comment | added | Neil Strickland | I like examples like $K=\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$. The Galois automorphisms are easy to exhibit and understand. The element $a=\sqrt{2}+\sqrt{3}+\sqrt{5}$ is not fixed by any automorphisms and so must generate the field; you can work out explicit representations of $\sqrt{2}$, $\sqrt{3}$ and $\sqrt{5}$ as polynomials in $a$. | |
| Dec 15, 2021 at 14:24 | comment | added | R.P. | Or that the formula for the $n$-th Fibonacci number is an integer (although that would of course also follow from the fact that its value is equal to the $n$-th Fibonacci number). | |
| Dec 15, 2021 at 14:22 | comment | added | R.P. | I think I would try to define the Galois group as the group (or set) of permutations of the set of roots that preserve the algebraic relations between them. But this is already pretty abstract. But you could give easy examples of irr. polynomials where there are nontrivial relations, so the group can be seen to be strictly smaller than $S_n$. An easy-to-explain corollary of Galois theory is that if an element is fixed under all element of the group, it us in the ground field. You could use this to prove that e.g. $(1+\sqrt{2})^n+(1-\sqrt{2})^n$ is an integer, and so on. | |
| Dec 15, 2021 at 13:31 | comment | added | Z. M | You could look for the book "Abel’s Theorem in Problems and Solutions", which assembles what V.I.Arnold's covers in a lecture to high school students. | |
| Dec 15, 2021 at 13:10 | history | asked | user929304 | CC BY-SA 4.0 |