abstract algebra - $K$ is a splitting field $\iff$ any irreducible ...
Let K/F K / F be a finite extension. I want to show that K K is a splitting field over F F any irreducible polynomial p(x) ∈ F[x] p (x) ∈ F [x] that has a root in K K splits completely over K K. …
If $f$ and $g$ are surjective, then $g (f (x))$ is surjective
Sep 25, 2016 · What part of the proof are you having trouble understanding? In my reading, the image you posted contains a complete and detailed proof directly from the definition of surjective.
Show that if $g \\circ f$ is injective, then so is $f$.
May 10, 2015 · The Problem: Let X, Y, Z X, Y, Z be sets and f: X → Y, g: Y → Z f: X → Y, g: Y → Z be functions. (a) Show that if g ∘ f g ∘ f is injective, then so is f f. (b) If g ∘ f g ∘ f is surjective, …
If - Mathematics Stack Exchange
I don't really follow your argument, but it seems like you're making this much more complicated than it is. If a graph is simple, then an edge is determined by a pair of vertices. Since there are …
abstract algebra - If $G/Z (G)$ is cyclic, then $G$ is abelian ...
@Robert: Yes, I think so. Where did the negative exponent come from? Would you want to make this comment a formal "answer"?
lie groups - Lie Algebra - Theory of prolongations, Criterion of ...
Apr 15, 2021 · I think I get what the Lie Algebra is all about (basically, you have a system of equations/diffeomorphisms (whose IFG are the elements of the Lie Algebra), and using the Lie …
discrete mathematics - Is "empty set" an element of a set ...
Sep 13, 2023 · The empty set is a subset of every set including itself, but it is not necessarily an element of any particular set.
If $φ:I→J$ is a homeomorphism then $f_n→f$ implies that
Jan 8, 2022 · If $φ:I→J$ is a homeomorphism then $f_n→f$ implies that $ (f_n∘φ) → (f∘φ)$ with respect the uniform, pointwise and $L_2$ topology respectively?
radicals - Is there a way to check if an integer is a square ...
Nov 11, 2019 · Is there a way to check if a number is square number? For example, we know that 4 4 is a square number because 22 = 2 2 2 = 2 and 9 9 is a square number because 32 = 9 3 …
Two Circles intersecting, common tangent: Proof $GI=IH$
Thanks for the link, but unfortunately I'm looking for a proof with triangles, congruences and so on.