Consider the following family of polynomials in $K[x,y]$, where $K$ has characteristic zero: $f_n(x,y)=(x+y)^n+(x-1)y^n,$ for $n\geq 3$. I can prove that $f_n(x,y)$ has an irreducible factor of ...
I'm trying to show that all possible splitting fields occur for a class of cubic polynomials, so have started by looking at the discriminants. Clearly, given that they are products of squares, all ...
Let $P_n(t)$ be polynomials with integer coefficients with $d_n = \deg(P_n(t))$ going to infinity when $n$ goes to infinity and with nonzero discriminants $disc(P_n(t)) \neq 0$. Question: Is $$ ...
This question is related to Degree of sum of algebraic numbers. Forgive me if this is a dumb question, but are there two algebraic numbers $a$ and $b$ of degree $3$ and $6$ respectively, such that the ...
[Lightly edited for copy and proper formatting of mathematics. -- Pete L. Clark] The Background: Let $T$ be a trapezoid. Sherman Stein, using valuation theory, showed that if $T$ is dissectible into ...
The following problem is my variant of something Irving Kaplansky noticed when we worked together. I do not think it is by nature a difficult problem, it is simply too hard for me to finish. Suppose ...