Simon Henry
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 Apr 24 awarded Enlightened Apr 24 awarded Nice Answer Apr 21 awarded Yearling Apr 21 comment Explicit Galois Action for $X^3 - X -1$ This is completely explicit : if you have any element of the field it can be written as a polynome in $x$ and $z$ and for any $\sigma \in \S_3$ you just replace all the $x$ and $z$ by their image under $\sigma$, exactly as you do in the two other examples you mentioned. Apr 21 comment Explicit Galois Action for $X^3 - X -1$ If by rational fonction you mean with rational coefficient then there cannot be : such a function would preserve the subfied generated by $x$ and this fields does not contains $z$. Apr 21 answered Explicit Galois Action for $X^3 - X -1$ Apr 15 answered Does “equalisers always closed” imply $T_2$? Dec 15 awarded Caucus Jan 25 comment Fibonacci numbers expressed as squares of lower Fibonacci numbers en.wikipedia.org/wiki/Fibonacci_number Dec 13 awarded Critic Oct 17 answered Eigenvalues of the “Laplacian” on [0,2$\pi]\subset\mathbb{R}$ Oct 13 comment Finite subgroups of $\operatorname{Aut}(\mathbb{P}^1)$ I think the third groupe of your line of isomorphism is in fact $SO(3,\mathbb{R})$, not $SO(3,\mathbb{C})$ and hence your problem is solved (you can check that $SO(3,\mathbb{C})$ doesn't have the good dimension as a real manifold if you're not covinced...). Oct 6 revised a set containing every limit points but not closed added 34 characters in body Oct 6 answered a set containing every limit points but not closed Oct 6 awarded Commentator Oct 6 comment Why are locally compact groups Weil complete? (I wrote as if the group was commutative, but that doesn't change anything if it is not, this will work in an arbitrary uniform space) Oct 6 comment Why are locally compact groups Weil complete? Exactly as you would do with a sequence on a metric space : You have $(u_i)$ a cauchy net and $(u_j)$ a subnet converging to $u$. Take $V$ a neighborhood of $u$, and $V'$ a neighborhood of $0$ such that if $(u-x)$ and $(x-a)$ are in $V'$ then $a$ is in $V$. write the cauchy propertie for $V'$, then write that there is therm of $(u_j)$ which is $V'$ close to $u$ for $j$ big enough to be converne bu the previous cauchy propertie you wrote, you can deduce from those two thing, that for any $i$ concerne bu the cauchy properties, $u_i$ is in $V$. Oct 5 answered Why are locally compact groups Weil complete? Oct 4 comment $A = B^2$ for which matrix $A$? by Starting to compute the Jourdan reduction of your matrix. you juste have to determine whether or not a matrix of the form $\lambda * I_n +N$ with $N$ a reduced nilpotent matrix has a square root or not. it seem that when $\lambda$ is non zero it's always the case. So in the end you have to find a way to know when a reduced nilpotent matrix has a square root or not, this seem to be a purely combinatorics question, and the answer will only depend on the dimension. (so the general answer will depend on the dimmension of the jordan block of the 0 eigenvalue) Oct 4 comment What if every function in $C(X)$ has finite spectrum? Well, if you have a net $(u_i)$ which does not converge to a value $x$. Then by (the negation of the) definition there exist a neighborhood $V$ of $u$ such that for every $i \in I$ there exist $j \in J$ such that $u_j$ is not in $V$. Hence the set of $i$ such that $u_i$ is not in $V$ is cofininal in $I$ and it gives us a sub net (the increasing function you're talking about is just the inclusion) of $u_i$ such that no "sub-sub-net" converge to $x$. which leads to a contradiction if the space is compact and if every converging subnet of $(u_i)$ converge to $u$.