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 $F(\alpha) =F(\alpha ^2)$ .

Give an example of a semigroup $S$ which has a left identity and in which every element has a right inverse but  $S$ is not a group.

Show that if $G$ is a group of order $2n$ , then there are exactly an odd number of elements of order $2$ in $G$ . Also show that if n is odd and $G$ is abelian, there is only one element of order $2$ .

Let $G$ be the group of all $2 \times 2$ matrices $\left( \begin{array}{cols} a&b \\ c&d \\ \end{array} \right)$, where $a,b,c,d$ are integers modulo $p,p$ a prime number, such that $ad - bc \neq 0$. Determine the order of $G$ .

Show that, in a group $G, \circ(ab) = \circ(ba)$ for all $a,b \in G$ . Further show that if $a$ and $b$ commute, and $\circ(a) =n_1 , \circ(b) =n_2$ and $(n_1,n_2) =1$, then $\circ(ab)= \circ(ba)=n_1 n_2$ .

Let $G$ be a group, $H \le G$ and $g \in G$ . Show that if $\circ(g) =n$ and $g^m \in H$, where $n$ and $m$ are coprime integers, then $g \in H$ .