1. Galois Theory!

Set F=F2(t) where t is a variable, let @ be a root of the equation x^2+x+t=0, set B=f(@) and set b=@^(1/2) and E=F(b). then B is separable over F and E is purely inseparable over B. show that there is no intermediate field B' such that E is separable over B' and B' purely inseparable over F.

Q2: Suppose that F is a field of characteristic p and that @ is a separable element in some extension. Show that then F(@^p)=F(@)

Any insights would be great!

2. Originally Posted by dabien
Set F=F2(t) where t is a variable, let @ be a root of the equation x^2+x+t=0, set B=f(@) (F@?)and set b=@^(1/2) and E=F(b). then B is separable over F and E is purely inseparable over B.
The deriative of $\displaystyle p(x)=x^2+x+t$ over $\displaystyle \mathbb{F}_2(t)$ is 1, which implies that the deriative has no root at all. Thus $\displaystyle p(x)=x^2+x+t$ is separable. It follows that $\displaystyle B=F(\alpha)$ is separable over F, where $\displaystyle \alpha$ is a root of p(x). Let $\displaystyle k(x)=x^2-\alpha \text{ } (=x^2+\alpha) \in B[x]$. We see that $\displaystyle k(x)$ is irreducible in $\displaystyle B[x]$. However, $\displaystyle k(x)$ is factored as $\displaystyle k(x)={(x - \sqrt{\alpha})}^2$ in $\displaystyle E[x]$. Thus E is purely inseparable over B.

show that there is no intermediate field B' such that E is separable over B' and B' purely inseparable over F.
[EDIT]

Q2: Suppose that F is a field of characteristic p and that @ is a separable element in some extension. Show that then F(@^p)=F(@)
Any insights would be great!
Observe that $\displaystyle \mathbb{F}^p=\mathbb{F}$, where $\displaystyle \mathbb{F} = F(\alpha)$ and p is a prime number.