Let $F$ be a field and $E = F(α)$ where $α^2 ∈ F$ and $α \notin F$. Determine the set $S := \lbrace β ∈ E|β^2 ∈ F\rbrace$

Question

Let $F$ be a field and $E = F(α)$ where $α^2 ∈ F$ and $α \notin F$. Determine the set $S := \lbrace β ∈ E|β^2 ∈ F\rbrace$.

How does one even go about starting such a question?

Answer 1

Since $\alpha ^2 \in F$ we know that $F \subset E$ is a degree two extension with basis $1, \alpha$. Thus any element can be written as $x+\alpha y$ where $x,y \in F$. Let us square one such typical element $(x+\alpha y)^2 = x^2+\alpha^2 y^2 +2 \alpha xy$ if this element is in $F$ then $xy$ must be equal to zero. Thus $S=F \cup \{ \alpha y : y \in F \}$ if the characteristic of $F$ is not 2, else $E=S$.

Answer 2

As a start, show that every element $\beta$ of $E$ can be written as $\beta= a+b\alpha$ with $a,b\in F$. Then, following the problem statement step by step, compute $\beta^2$, taking into account that $\alpha^2\in F$, and check what condition you find for $\beta^2$ to be in $F$.

Remark: Be careful that the answer depends on the characteristic of $F$.