Tag Archives: alternating group

Symmetrical polynomial under even permutation

$\mathbf{Problem:}$ Let $F$ be of characteristic $\neq 2$ and also let $H$ be the subgroup of $G=\text{Gal }E/F(p_1,p_2,\cdots,p_n)$ corresponding to the alternating group, that is the set of $\zeta(\pi)$, $\pi\in A_n$. Show that $\text{Inv}H=F(p_1,p_2,\cdots,p_n,\Delta)$ where $\displaystyle\Delta=\prod\limits_{i.

$\mathbf{Proof:}$ It is a basic fact that $\zeta(\pi)(\Delta)=\Delta$ when $\pi$ is an even permutation and $\zeta(\pi)(\Delta)=-\Delta$ when $\pi$ is an odd permutation.

So all $\pi\in H$ preserve $\Delta$ and $p_1,p_2,\cdots,p_n$, which means $F(p_1,p_2,\cdots,p_n,\Delta)\subset \text{Inv}H$.

On the other hand no odd permutation can preserve $\Delta$, then $\text{Gal }E/F(p_1,p_2,\cdots,p_n,\Delta)\leq H$.

Combining these two, we know $\text{Inv}H=F(p_1,p_2,\cdots,p_n,\Delta)$.

$\text{Q.E.D}\hfill \square$

$\mathbf{Remark:}$ Jacobson p244.