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Let $x_1,x_2,x_3,\ldots,x_n$ be the roots of a polynomial $P_n(x)$. Let $F$ be the field $\mathbb{Q}[x_1,x_2,x_3,\ldots,x_n]$, i. e. all the possible combinations of rational numbers with $x$'s. It's ...

The discriminant $\Delta(P)$ of a monic polynomial $P(x)=x^n + a_{n-1} x^{n-1} + \dotsb + a_0$ of degree $n$, when expanded (using elementary symmetric polynomials), is a symmetric polynomial of ...

In Edwards' "Galois Theory" articles 29-31, the notion of Galois resolvent is motivated by a result of Lagrange (article 104 in his Réflexions sur la résolution algébrique des équations). ...

Consider a motivating example: Let $E\in \mathbb{Q}[y][x]$ be of degree $n=2$ (in $x$) and separable when viewed as a member of $\mathbb{Q}[x,y]$. Therefore we can calculate it's roots in $\mathbb{Q}[...