Let \(R\) be an integral domain. Prove that every finitely generated torsion \(R\)-module has a non-zero annihilator. Give an example of a torsion \(R\)-module whose annihilator is the zero ideal

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 \(f(x)\) .