Tutorial 2 : Symmetric Polynomials and Splitting Fields


1. Let $ F$ be a field and % latex2html id marker 28101
$ f(x)=\prod_{i=1}^n(x-r_i)$ where $ r_1, r_2 \ldots, r_n \in \overline {F}.$ Show that
$\displaystyle \:$disc% latex2html id marker 28106
$\displaystyle \;(f(x))=(-1)^{\binom{n}{2}}\prod_{i=1}^nf'(r_i).$


2. Show that $ \:$disc$ \;(x^4+px^2+q)=16q(p^2-4q)^2.$



3. (a)Show that $ \displaystyle (x^n-1)=(-1)^{\binom{n}{2}+n-1}n^n.$
    (b)
Let $ g(x)$ and $ h(x)$ be monic polynomials and $ g(x)=(x-a)h(x).$ Show that         $ \displaystyle (g(x))=h(a)^2dis(h(x)).$
     (c)
Show that $ \displaystyle (x^{n-1}+ x^{n-2}+ \cdots +1)=(-1)^{(n-1)(n+2)/2}n^{n-2}.$




4. Show that a polynomial $ f \in S:=R[x_1, x_2, \ldots, x_n]$ where $ R$ is a commutative ring having $ 2^{-1}$ is fixed under all the automorphisms of $ S$ induced by even permutations in $ S_n$ if and only if $ f=g+\delta h$ where $ g$ and $ h$ are symmetric polynomials and % latex2html id marker 28222
$ \delta= \prod_{i \leq j}(x_i -x_j).$


5. Find the splitting field of $ x^6+x^3+1.$




6. Let $ K \subset \mathbb{C}$ be a splitting field of $ f(x)=x^3-2$ over $ \mathbb{Q}.$ Find a complex number $ z$ such that $ K=\mathbb{Q}(z).$



7. Let $ F$ be a field of characteristic $ p.$ Let $ f(x)=x^p-x-c \in F[x].$ Show that either all roots of $ f(x)$ lie in $ F$ or $ f(x)$ is irreducible in $ F[x].$



8. Let $ E$ be a splitting field over a field $ F$ of $ f(x).$ Let $ K$ be a subfield of the field extension $ E/F.$ Let $ \sigma : K \rightarrow E$ be an $ F$-embedding. Show that $ \sigma$ can be extended to an automorphism of $ E.$



9. Find a splitting field $ K$ of $ x^3-10$ over $ \mathbb{Q}(\sqrt{2}).$ Find $ [K:\mathbb{Q}].$