Let $F$ be a field and let $f(x)\in F[x]$. Denote the splitting field of $f(x)$ by $E$. Denote the roots of $f(x)$ by $\alpha_1,...,\alpha_n$. A polynomial $g(x_1,...,x_n) ∈ F[x_1,...,x_n]$ gives an relation between the roots if $g(α_1,...,α_n) = 0.$ Now let $σ\in Aut_F(E)$, and let $τ$ denote...

Let F be a field. Let $f(x) ∈ F[x]$ be a nonzero polynomial, let $E$ be a splitting field for $f(x)$ over $F$, and let $X$ denote the set of roots of $f(x)$ in E Let $f(x)\in F[x]$, and suppose $f(x) = f_1(x)···f_r(x)$ is the factorization of $f(x)$ into irreducible polynomials in $F[x]$. For each...

I know that the higher homotopy groups $\pi_n(X)$ can be defined in two way. Homotopy classes of the continuos maps $(I^n,\partial I^n)\longrightarrow (X,x_0)$. Homotopy classes of the continuos maps $(S^n,s_0)\longrightarrow (X,x_0)$. These two ways to define the higher homotopy groups are e...

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!

I have just created a betting → gambling synonym, and merged the former into the latter. I'm not exactly sold on the usefulness of either tag, but for the time being I guess they are basically talking about the same thing.

People always like to evaluate the variance, but is there any way for variance to be interesting to the gambling game makers? In another word, what is a pratical gambling game that involving some distributions that is relating to variance other than the normal distribution?

The completeness theorem fails for second-order logic. This question has some nice examples of consistent second-order theories without models. But non of them is finitely axiomatizable, at least those examples use infinitely many axioms. Are there consistent finitely axiomatizable second-ord...