john mangual
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 Apr21 comment Explicit Galois Action for $X^3 - X -1$ @kconrad I took that example from your note. maybe you meant something else Apr21 revised How to write $\sqrt{4x^2 - 3}$ in the ring $\mathbb{Q}[x]/(x^3 - x - 1)$? added 256 characters in body Apr21 comment Explicit Galois Action for $X^3 - X -1$ 1 OK. This extension is not Galois since $\mathrm{Aut}[\mathbb{Q}[x]/\mathbb{Q}]= 6 > [\mathbb{Q}[x]:\mathbb{Q}]=3$ $$.$$ 2 The action of the Galois group of $x^3 - x - 1$ cannot be extended to all of $\mathbb{C}$ but maybe it can be done using Riemann surfaces? The Riemann surface for $y^2 = x^2 + ax + b$ is always a sphere $\hat{\mathbb{C}}$ but for $y^2 = x^3 + ax + b$ it is hyperelliptic curve. Maybe the Galois action can be extended to that object. Apr21 asked How to write $\sqrt{4x^2 - 3}$ in the ring $\mathbb{Q}[x]/(x^3 - x - 1)$? Apr21 comment Explicit Galois Action for $X^3 - X -1$ With a formula you can try to insert values outside of the field $\mathbb{Q}[x]$. In the quadratic case $x^2 + ax + b$ you can write either $x \mapsto -(a + x)$ or $x \mapsto \frac{b}{x}$. These maps can be defined over $\mathbb{C}$ and they are involutions, wherever they are defined. It may be too much to hope for a similar situation for the cubic. Apr21 comment Explicit Galois Action for $X^3 - X -1$ This is not very explicit at all. Is there a rational substitution $x \mapsto \frac{p(x)}{q(x)}$ such that $x \leftrightarrow z$ ? Apr21 asked Explicit Galois Action for $X^3 - X -1$ Apr20 comment Prove that $\mathcal{O}_3$ and $\mathcal{O}_7$ are euclidean domains See also: Keith Conrad Euclidean Domains Apr20 revised Prove that $\mathcal{O}_3$ and $\mathcal{O}_7$ are euclidean domains erroneous figure replaced Apr20 answered Prove that $\mathcal{O}_3$ and $\mathcal{O}_7$ are euclidean domains Apr14 comment What is $\left[\frac{1}{2}(p-1)\right]! \;(\text{mod } p)$ for $p = 4k+1$? @nayrb A similar statement is Gauss' Lemma in Number Theory which counts the number of residues of $\{ a, 2a, \dots, \tfrac{p-1}{2} a\}$ less than $p/2$. Apr14 comment What is $\left[\frac{1}{2}(p-1)\right]! \;(\text{mod } p)$ for $p = 4k+1$? @nayrb the original theorem just asks about whether $\nu$ is even or odd. that might be good enough. Apr14 revised What is $\left[\frac{1}{2}(p-1)\right]! \;(\text{mod } p)$ for $p = 4k+1$? added 218 characters in body Apr14 revised What is $\left[\frac{1}{2}(p-1)\right]! \;(\text{mod } p)$ for $p = 4k+1$? deleted 26 characters in body; edited title Apr14 comment What is $\left[\frac{1}{2}(p-1)\right]! \;(\text{mod } p)$ for $p = 4k+1$? whoops :-) my 2nd question would be nontrivial if I said "less than $\frac{p}{4}$" Apr14 asked What is $\left[\frac{1}{2}(p-1)\right]! \;(\text{mod } p)$ for $p = 4k+1$? Apr14 asked Continued fraction for $[1,2,3,4,5,6,\dots]$ Apr13 awarded Necromancer Apr8 revised How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n$ as $n\to\infty$? defective link Apr8 comment Fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$ Another version of this discussion appears in Section 2.7 of Borevich + Shafarevich number theory book (PDF) in the Chapter on "Decomposable Forms", meaning that $x^2 - dy^2 = (x - \sqrt{d} y)(x + \sqrt{d}y)$ has complete factorization. Pell's equation can be thought of as a special case of Dirichlet Unit Theorem in Section 2.4 or part of Quadratic Forms in Section 2.7