Let $K/F$ be an extension of fields. Show that if every ring $R$ with $F \subseteq R \subseteq K$ is a field, then $K$ is an algebraic extension of $F$ .

Find the degree and a basis for $\mathbb{Q}( \sqrt{2} , \sqrt{3} )$ over $\mathbb{Q}$ .Also show that $\mathbb{Q}( \sqrt{2} , \sqrt{3} ) = \mathbb{Q}( \sqrt{2} + \sqrt{3} )$

Find the degree and a basis for $\mathbb{Q} ( \sqrt{3} + \sqrt{5} )$ over $\mathbb{Q} ( \sqrt{15})$ .

Prove that $[ \mathbb{Q} ( \sqrt {3-4i} \; + \; \sqrt{3 + 4i} ) : \mathbb{Q} ]=1$ .

Determine the degree of the extension  $\mathbb{Q} \large ( \sqrt {1 +\sqrt{-3}} \; + \sqrt{1- \sqrt{-3}} )$ over $\mathbb{Q}$ .

Give an example of a field $F$ and elements $a$ and $b$ from some extension field such that $F(a,b) \neq F(a)$ and $F(a,b) \neq F(b)$ , and $[F(a,b):F] < [F(a) :F][f(b):F]$ .

Determine the splitting field $E$ for each of the following polynomials over $\mathbb{Q}$ . Also determine $[E: \mathbb{Q}]$ in each case.

(i) $x^4 -2$    (ii) $x^4 -6x^2 -7$    (iii) $x^5 -3x^3 + x^2 -3$ .

Determine the minimal polynomial of $\sqrt{-3} \; + \sqrt{2}$ over $\mathbb{Q}$ .

Let $f(x)$ be any polynomial over $F$ and $\alpha$ be any automorphism of a field $k \supseteq F$ such that $\alpha$ leaves every element of $F$ fixed, i.e., $\alpha(a) =a \forall a \in F$. Show that for any root $\alpha$ of $f(x)$ in $K, \sigma(\alpha)$ is also a root of $f(x)$ .

Let $m$ be a positive integer, which is not a perfect square. Show that the mapping $\alpha + \beta \sqrt{m} \mapsto \alpha - \beta \sqrt{m}, \alpha ,\beta \in \mathbb{Q}$ , is an automorphism of $\mathbb{Q} (\sqrt{m})$ . Hence show that for any rational numbers $\alpha , \beta$ with $\beta \neq 0$ , the minimal polynomials of $\alpha + \beta \sqrt{m}$ and $\alpha - \beta \sqrt{m}$ over $\mathbb{Q}$ are same .

Let $F$ be a field and $\alpha$ is algebraic over $F$. Show that if $[F(\alpha):F)]$ is odd, then $F(\alpha) =F(\alpha ^2)$ .