1. The polynomail $\displaystyle f(x) = x^5-5x+1$ is irreducible in $\displaystyle \mathbb Q [x]$. Find the number if real roots of $\displaystyle f(x).$

2. Let M be the splitting field for $\displaystyle f(x)$ over $\displaystyle \mathbb Q.$ Show that $\displaystyle M/\mathbb Q$ is a Galois extension

and determine its Galois group $\displaystyle Gal(M/\mathbb Q).$ Is $\displaystyle f(x)$ solvable by root?

In the following it can be used without a proof that the Discrimentant of $\displaystyle f(x)$

is $\displaystyle -3\cdot5^6\cdot 17.$

3. Show that $\displaystyle \mathbb Q(\sqrt{-51})$ is contained in $\displaystyle M$ and determine that $\displaystyle f(x)$

is irreducible in $\displaystyle \mathbb Q(\sqrt{-51})[x].$

4. Determine the Galois group $\displaystyle Gal(M/ \mathbb Q(\sqrt{-51})).$