Let us consider the polynomial $f(x)=x^3+x^2-4x+1$. I was asked the following things:
(1) Prove that $f(x)$ has one and only one negative root.
For this I just used Bolzano's theorem and noticed it has one negative and two positive real roots.
(2) Is $f(x)$ irreducible over $\mathbb Q$?
I just checked it has no roots mod $2$.
(3) Prove that $\tau(\alpha)=1/(1-\alpha)$ is automorphism of $\mathbb Q(\alpha)$. (Here $\alpha$ is the negative root which exsistence we proved in (1)).
Here I did a pretty long calculations to check that $\tau$ really is automorphism. Clearly $\mathbb Q(\alpha)=\{c_0+c_1 \alpha+c_2 \alpha^2 |c_i \in \mathbb Q\}$. So I took $a,b \in \mathbb Q(\alpha)$ and calculated
$$ \tau(a+b)=\tau(a_0+a_1\alpha+a_2\alpha^2+b_0+b_1\alpha+b_2\alpha^2)=a_0+a_1\tau(\alpha)+a_2\tau(\alpha)^2+b_0+b_1\tau(\alpha)+b_2\tau(\alpha)^2= \cdots = \tau(a)+\tau(b). $$ And also that $$ \tau(ab)=\tau(a)\tau(b) $$
(4) Is $\mathbb Q(\alpha)/\mathbb Q$ a normal extension?
I know that it is normal if it contains the remaining two positive real roots of the $f(x)$. However I have no idea how to prove/disaprove that since I dont know the roots explicitely.
So is my work correct so far and how to deal with the last question?