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Apr
21
comment Explicit Galois Action for $X^3 - X -1$
... with the caveat that only the $A_3 \subseteq S_3$ subgroup of permutations can be written in terms of substitution $x \mapsto p(x)$. (that is, you only get the Galois group of $\mathbb{Q}(x,z) / \mathbb{Q}(\sqrt{-23}))$.
Apr
21
comment Explicit Galois Action for $X^3 - X -1$
(ah, two comments up, that last thing should have been $r(x,z) \mapsto r(z, -x-z)$)
Apr
21
comment Explicit Galois Action for $X^3 - X -1$
(and, at least in this example, the action can be lifted to a group of automorphisms on the rational function field $\mathbb{Q}(X,Z)$, which I think is typically possible, and even the polynomial ring $\mathbb{Q}[X,Z]$, which I think is typically not possible)
Apr
21
comment Explicit Galois Action for $X^3 - X -1$
@john: Except in very rare circumstances, field automorphisms do not act like plugging a field element into a rational function. Instead, in many common situations it's exactly the other way around: you want to view your field elements as rational functions, and then the Galois action is to evaluate them. e.g. here, any permutation of $\{ x, z, -x-z \}$ corresponds to a Galois action, e.g. the action $r(x,y) \mapsto r(z, -x-z)$.
Apr
21
comment Explicit Galois Action for $X^3 - X -1$
The identity morphism is the only automorphism of $\mathbb{Q}(x)$. If you want to consider nontrivial automorphisms of a field containing $x$ that do not fix $x$, you must consider, at a minimum, $\mathbb{Q}(x,z)$.
Apr
19
comment Polynomial approximation on affine varieties
@tj_: I fail to see how a proof that the answer is yes (when $V \cap W = \varnothing$) counts as nitpicking and fails to be an answer.
Apr
19
answered Polynomial approximation on affine varieties
Apr
15
comment Norm of ideals in arbitrary order of number field
A suggested edit substantially changed the mathematical comment of this answer. $I$ is quite deliberately the principal ideal of $\mathbb{Z}[3i]$ generated by $3i$, not the principal ideal of $\mathbb{Z}[i]$ generated by $3i$.
Apr
15
revised Norm of ideals in arbitrary order of number field
rolled back to a previous revision
Apr
15
awarded  Great Answer
Apr
7
comment Differential Forms Notation is Wrong? Confirm or deny?
I don't think anything in your post contradicts the reason why I like differential forms: I believe partial derivatives are an awkward formulation of multivariable calculus, and the notation for partial derivatives is terrible.
Apr
7
reviewed Leave Closed Differential Forms Notation is Wrong? Confirm or deny?
Apr
7
reviewed Leave Open If $\operatorname{trace}(Z)=0$, then there exist $X$, $Y$, $|Y|\not=0$ satisfy $Z=XY-YX$
Apr
7
reviewed Close Proving that $f(x) \le\ g(x)$, for all $x \in\ [c,b)$
Apr
7
reviewed Leave Open Given $2015$ points, show that it is possible to separate them such that $1007$ of them lie inside the circle
Apr
7
reviewed Close Determine E(X) of X Where X Is Number Of Days Beer Is Drank On The Same Day?
Apr
7
reviewed Close Let $R$ be a commutative ring with $1$. Prove that $R[X,Y] = R[X][Y]$.
Apr
7
comment Goldbach's conjecture is wrong?!
@Prime: This isn't like a message forum; you can't really use answers as a "reply" to other answers.
Apr
7
awarded  Nice Answer
Apr
7
comment Goldbach's conjecture is wrong?!
@Donkey: The OP seems to be claiming that for sufficiently large numbers, there aren't primes with a small gap between them. But it's wholly unclear how this fits into his argument (which is itself unclear)