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 \(F(\alpha) =F(\alpha ^2)\) .