john mangual
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 Apr 23 comment modification of Dedekind cuts I don't know if my model accurately describes continued fractions, but you certainly answered my question. Apr 23 comment modification of Dedekind cuts In calculus there is $dt^2 = 0$. Here if $\epsilon=1/n$ the ring should be $\mathbb{R}[\epsilon]/(\epsilon^3)$ Apr 23 revised modification of Dedekind cuts added 182 characters in body Apr 23 asked modification of Dedekind cuts Apr 23 answered Algebraic number theory topics for undergrads Apr 22 answered What book or website has nice, colorful diagrams illustrating real quadratic integer rings? Apr 22 accepted Explicit Galois Action for $X^3 - X -1$ Apr 22 revised 3-cycles of the iterated quadratic maps and the Galois Theory of $x^3 = a(x-1)$ more appropriate title Apr 22 revised 3-cycles of the iterated quadratic maps and the Galois Theory of $x^3 = a(x-1)$ added 1 character in body Apr 22 asked 3-cycles of the iterated quadratic maps and the Galois Theory of $x^3 = a(x-1)$ Apr 22 comment Explicit Galois Action for $X^3 - X -1$ Looks like my quadratic equation proof is correct except I forgot to adjoin the square root of the discriminant - and then it matches your Theorem 2.6 The examples on Table 2 have lots of $A_3$ examples too. Apr 22 revised Explicit Galois Action for $X^3 - X -1$ added 137 characters in body Apr 21 revised Explicit Galois Action for $X^3 - X -1$ added 281 characters in body Apr 21 comment Explicit Galois Action for $X^3 - X -1$ @kconrad I learned today the spitting field can be bigger than adjoining the element. I think the substitution I'm looking for stems from the action of the Galois group on $\mathbb{Q}(x, \sqrt{-23})$ on the basis generated by powers of $x$ and the square root of -23 Apr 21 comment Explicit Galois Action for $X^3 - X -1$ @kconrad I took that example from your note. maybe you meant something else Apr 21 revised How to write $\sqrt{4x^2 - 3}$ in the ring $\mathbb{Q}[x]/(x^3 - x - 1)$? added 256 characters in body Apr 21 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. Apr 21 asked How to write $\sqrt{4x^2 - 3}$ in the ring $\mathbb{Q}[x]/(x^3 - x - 1)$? Apr 21 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. Apr 21 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 $x \leftrightarrow z$ ?