Let \(K/F\) be an extension of fields. Let \(\alpha ,\beta \in K\) be algebraic over \(F\) and the degrees of the minimal polynomials of \(\alpha\) and \(\beta\) over \(F\) are \(m\) and \(n\), respectively. Show that if \((m,n) =1\), then \([F(\alpha ,\beta ):F] =mn\) .

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 .