2 Problem Sheet 2:
Field extensions and minimal polynomials
Exercise 2.1 By considering degrees of field extensions, determine which of the following numbers are algebraic (over \(\mathbb{Q}\)).
\(\;(a)\;\;1+\sqrt{3}+\sqrt{5},\hspace{2em}\) \(\;(b)\;\;\sqrt{2}+\sqrt[4]{2},\hspace{2em}\) \(\;(c)\;\;\sqrt{2}+\sqrt[3]{3},\)
\(\;(d)\;\;\sqrt{\pi}-1,\hspace{4.4em}\) \(\;(e)\;\;\sqrt{\pi}+\sqrt{7},\hspace{1.9em}\) \(\;(f)\;\;\sqrt[5]{1+\sqrt{e}}.\)
Exercise 2.2 Find the minimal polynomial of \(\theta\) over the field \(K\) where:
\(\;(a)\;\;\theta=\sqrt[3]{1+\sqrt{3}}\) and \(K=\mathbb{Q},\hspace{2em}\) \(\;(b)\;\;\theta=\sqrt{2}+\sqrt[3]{2}\) and \(K=\mathbb{Q},\)
\(\;(c)\;\;\theta=e^{2\pi i/5}\) and \(K=\mathbb{Q},\hspace{3.75em}\) \(\;(d)\;\;\theta=\sqrt{2}\) and \(K=\mathbb{Q}(i),\)
\(\;(e)\;\;\theta=\sqrt[3]{2}\) and \(K=\mathbb{Q}(\sqrt{2}).\)
Exercise 2.3 For the following field extensions \(L/K\), find a basis of \(L\) (as a vector space over \(K\)).
\(\;(a)\;\;L=\mathbb{Q}(\sqrt{2}\,,\,i)\) over \(K=\mathbb{Q},\hspace{5.1em}\) \(\;(b)\;\;L=\mathbb{Q}(\sqrt{2}\,,\,\sqrt{5})\) over \(K=\mathbb{Q},\)
\(\;(c)\;\;L=\mathbb{Q}(\sqrt[3]{5}\,,\,i)\) over \(K=\mathbb{Q},\hspace{5.1em}\) \(\;(d)\;\;L=\mathbb{Q}(\sqrt[3]{2}-2\sqrt[3]{4})\) over \(K=\mathbb{Q},\)
\(\;(e)\;\;L=\mathbb{Q}(\sqrt{2}\,,\,\sqrt{3}\,,\,\sqrt{7})\) over \(K=\mathbb{Q},\hspace{2em}\) \(\;(f)\;\;L=\mathbb{Q}(\sqrt[3]{1+\sqrt{3}})\) over \(K=\mathbb{Q}(\sqrt{3}),\)
\(\;(g)\;\;L=\mathbb{R}(x)\) over \(K=\mathbb{R}(x+x^{-1}),\hspace{3em}\) \(\;(h)\;\;L=\mathbb{R}(x)\) over \(K=\mathbb{R}(x^2+x^{-2}),\)
\(\;(i)\;\;L=\mathbb{F}_2(\alpha)\) over \(K=\mathbb{F}_2\) where \(\alpha^4+\alpha+1=0,\)
\(\;(j)\;\;L=\mathbb{F}_3(\alpha)\) over \(K=\mathbb{F}_3\) where \(\alpha^3+\alpha^2+2=0.\)
Exercise 2.4 Write each of the following in the form \(a+b\sqrt[3]{2}+c\sqrt[3]{4}\) with \(a, b, c\in\mathbb{Q}\):
\(\;(a)\;\;\dfrac{1}{\sqrt[3]{2}},\hspace{2em}\) \(\;(b)\;\;\dfrac{1}{1+\sqrt[3]{2}},\hspace{2em}\) \(\;(c)\;\;\dfrac{1+\sqrt[3]{2}}{1+\sqrt[3]{2}+\sqrt[3]{4}},\)
\(\;(d)\;(\star)\;\dfrac{1}{r+s\sqrt[3]{2}+t\sqrt[3]{4}}\;\;\) for arbitrary \(r,s,t\in\mathbb{Q}\) with \(r+s\sqrt[3]{2}+t\sqrt[3]{4}\neq 0\).
Exercise 2.5 Write each of the following in the form \(a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}\) with \(a, b, c, d\in\mathbb{Q}\):
\(\;(a)\;\;\dfrac{1}{\sqrt{2}+\sqrt{3}},\hspace{2em}\) \(\;(b)\;\;\dfrac{1}{1+\sqrt{2}+\sqrt{3}},\hspace{2em}\) \(\;(c)\;\;\dfrac{\sqrt{2}+\sqrt{3}}{1+\sqrt{2}+\sqrt{3}+\sqrt{6}}.\)
Exercise 2.6 Let \(L=\mathbb{F}_2(\theta)\) where \(\theta\) is a root of \(f(x)=x^4+x+1\in\mathbb{F}_2[x]\). Write each of the following in the form \(a+b\theta+c\theta^2+d\theta^3\) with \(a, b, c, d\in\mathbb{F}_2\):
\(\;(a)\;\;\theta^{-1},\hspace{3em}\) \(\;(b)\;\;\theta^5,\hspace{3em}\) \(\;(c)\;\;\theta^{15},\hspace{3em}\) \(\;(d)\;\;(1+\theta+\theta^2)^{-1}.\)
Exercise 2.7 Let \(L=\mathbb{F}_2(\theta)\) where \(\theta\) is a root of \(f(x)=x^3+x^2+1\in\mathbb{F}_2[x]\). Show that \(f(x)\) can be written as a product of three linear factors with coefficients in \(L\).
Exercise 2.8 Find the minimal polynomial of \(\alpha=\sqrt{7}+i\in\mathbb{C}\) over the field \(K\) in the following cases:
\(\;(a)\;\;K=\mathbb{R},\hspace{3em}\) \(\;(b)\;\;K=\mathbb{Q}(i),\hspace{3em}\) \(\;(c)\;\;K=\mathbb{Q}.\)
Exercise 2.9 Find the minimal polynomials over \(\mathbb{Q}\) of the following algebraic numbers:
\(\;(a)\;\;\sqrt{1+\sqrt{2}},\hspace{3em}\) \(\;(b)\;\;\sqrt{\sqrt{2}-i}.\)
Exercise 2.10 Find the degree \([L:K]\) of the following field extensions and give a basis for \(L\) over \(K\):
\(\;(a)\;\;L=\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})\) over \(K=\mathbb{Q},\hspace{3.75em}\) \(\;(b)\;\;L=\mathbb{Q}(\sqrt{2},\sqrt[3]{2})\) over \(K=\mathbb{Q},\)
\(\;(c)(\star)\;\;L=\mathbb{Q}(\sqrt[3]{2},\sqrt[3]{6},\sqrt[3]{24})\) over \(K=\mathbb{Q},\hspace{2em}\) \(\;(d)\;\;L=\mathbb{Q}(\sqrt{2},\sqrt{3})\) over \(K=\mathbb{Q}(\sqrt{2}+\sqrt{3}),\)
\(\;(e)\;\;L=\mathbb{Q}(\sqrt{2},\sqrt{6}+\sqrt{10})\) over \(K=\mathbb{Q}(\sqrt{3}+\sqrt{5}).\)
Exercise 2.11 Prove that \(f(x)=x^2-5\) is irreducible over \(\mathbb{Q}(\sqrt[3]{7})\).
Exercise 2.12 \((\star)\;\) Let \(p>2\) be an odd prime. Show that there is exactly one field (up to isomorphism) with \(p^2\) elements. Hint: try completing the square.