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 Yearling
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Nov
4
comment Checking irreducibility of a polynomial in $\mathbb{K}[x,y]$ and PAC fields
To be separable in my post means to have no multiple roots in the algebraic closure.
Oct
31
answered Checking irreducibility of a polynomial in $\mathbb{K}[x,y]$ and PAC fields
Oct
26
comment Seeking a proof that the residual field of the decomposition field is equal to the base residual field
It is true that in general $K\subseteq Z$ is not Galois. But one does not need that: all the computations are taking place within $L$.
Oct
26
comment Proving that a Finite Field Over Its Prime Field Is Galois
I agree with Andy Tam: Lagrange shows that $f$ has $p^n$ distinct roots in $E$ and its degree is $p^n$. Hence it must be separable.
Oct
25
comment Prove there is a natural isomorphism between $L(V,L(V,W))$ and $Bil(V \times V,W) $.
Yes. Before you can talk about the linearity of a map, you have to make sure that it maps a vector space to a vector space.
Oct
23
revised Prove there is a natural isomorphism between $L(V,L(V,W))$ and $Bil(V \times V,W) $.
added 443 characters in body
Oct
23
comment Seeking a proof that the residual field of the decomposition field is equal to the base residual field
I don't remember where I saw this proof for the first time. Probably in a seminar on Valuation Theory given by Franz-Viktor Kuhlmann in the 1980s. He also told me that many existing proofs, although following essentially the same line of arguments, are just too complicated at one or the other point -- see for example the proof in Zariski-Samuel, volume 2. Anyway there should also be a simple proof in Abhyankar's book "Ramification theoretic methods in Algebraic Geometry".
Oct
23
revised Prove there is a natural isomorphism between $L(V,L(V,W))$ and $Bil(V \times V,W) $.
added 336 characters in body
Oct
23
comment Prove there is a natural isomorphism between $L(V,L(V,W))$ and $Bil(V \times V,W) $.
I will expand my post.
Oct
23
comment Prove there is a natural isomorphism between $L(V,L(V,W))$ and $Bil(V \times V,W) $.
Yes. $b(v,\cdot)$ is the linear map ones gets through giving the first variable of $b$ the fixed value $v$. In other words: $b$ is considered as a parametrized family of linear maps, that depend linearly on the parameter.
Oct
23
answered Prove there is a natural isomorphism between $L(V,L(V,W))$ and $Bil(V \times V,W) $.
Oct
23
answered Seeking a proof that the residual field of the decomposition field is equal to the base residual field
Oct
22
answered Is $\mathbb Q$ a metric space?
Oct
21
answered if $(x_0,y_0)$ is local extrema in $ax^2 + by^2 + cxy + dx + ey + l$ then its global too.
Oct
20
answered How to express cycle notation with a cyclic group?
Oct
16
answered Show that it could be that $[LK:K] \lt [L: L\cap K]$
Oct
13
awarded  Yearling
Oct
9
comment Proving Cayley's Theorem
Let $f:A\rightarrow B$ and $g:B\rightarrow C$ be two maps. Then their composition $g\circ f$ is the map $A\rightarrow C$ defined by $(g\circ f)(a):=g(f(a))$.
Oct
9
comment Proving Cayley's Theorem
$\mathrm{Sym}(G)$ is the set of all permutations, that is bijections $G\rightarrow G$, together with the composition of maps.
Oct
6
answered How to profile people using clustering