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 \(g \in H\) .