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2d
accepted modification of Dedekind cuts
2d
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.