$$ \DeclareMathOperator*{\Hom}{Hom} \DeclareMathOperator{\Map}{Map} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\ker}{ker} \DeclareMathOperator{\ch}{char} \DeclareMathOperator{\Frac}{Frac} \DeclareMathOperator*{\Gal}{Gal} $$

3 Problem Sheet 3:
Properties of field extensions

Exercise 3.1 Find a basis of the splitting field \(L\) of \(f(x)\) over \(K\) in the following cases:

\(\;(a)\;\;f(x)=(x^2-2)(x^2-5)\) over \(K=\mathbb{Q},\hspace{2em}\) \(\;(b)\;\;f(x)=x^4+1\) over \(K=\mathbb{Q},\)

\(\;(c)\;\;f(x)=x^8-2\) over \(K=\mathbb{Q},\hspace{6.15em}\) \(\;(d)(\star)\;\;f(x)=x^4+x^2-1\) over \(K=\mathbb{Q},\)

\(\;(e)\;\;f(x)=x^6-2\) over \(K=\mathbb{Q}(\sqrt{-3}),\hspace{3.15em}\) \(\;(f)\;\;f(x)=x^8-2\) over \(K=\mathbb{Q}(i).\)


Exercise 3.2 Let \(\theta\) be a root of the polynomial \(f(x)=x^3+x+1\in\mathbb{F}_2[x]\). Show that the extension \(\mathbb{F}_2(\theta)\) is a splitting field for \(f(x)\) over \(\mathbb{F}_2\).


Exercise 3.3 Let \(p\) be a prime number. What is the degree of the splitting field of \(x^p-1\) over \(\mathbb{Q}\)?


Exercise 3.4 Decide whether or not the following pairs of fields are isomorphic:

\(\;(a)\;\;\mathbb{Q}(\sqrt[4]{2})\) and \(\mathbb{Q}(i\sqrt[4]{2}),\hspace{5em}\) \(\;(b)\;\;\mathbb{Q}(\sqrt[3]{1+\sqrt{3}})\) and \(\mathbb{Q}(\sqrt[3]{1-\sqrt{3}}),\)

\(\;(c)\;\;\mathbb{Q}(\sqrt{2})\) and \(\mathbb{Q}(\sqrt{3}),\hspace{5.4em}\) \(\;(d)\;\;\mathbb{Q}(\pi)\) and \(\mathbb{Q}(e)\).


Exercise 3.5 Describe the splitting field of \(x^3-11\) over \(\mathbb{Q}\). Show that if \(\alpha\) is any root of this polynomial then \(\mathbb{Q}(\alpha)/\mathbb{Q}\) is not normal.


Exercise 3.6 Suppose that \(K\subset M\subset L\) is a tower of finite extensions. For each of the following, either prove the statement or provide a counterexample.

\(\;(a)\;\) If \(L/M\) and \(M/K\) are both normal, then \(L/K\) is normal.

\(\;(b)\;\) If \(L/K\) is normal, then \(M/K\) is normal.

\(\;(c)\;\) If \(L/K\) is normal, then \(L/M\) is normal.


Exercise 3.7 Are the following true or false? Justify your answers.

\(\;(a)\;\) A degree \(2\) extension \(L/K\) is normal.

\(\;(b)\;\) A degree \(2\) extension \(L/K\) is separable.


Exercise 3.8 Suppose \(K\subset M\subset L\) be a tower of finite extensions.

\((a)\) Show that if \(L/K\) is separable, then both \(L/M\) and \(M/K\) are separable.

\((b)(\star)\) Show that if \(L/M\) and \(M/K\) are separable, then \(L/K\) is separable.

(For \((b)\), you may use the following characterisation of separable extensions in characteristic \(p\): A finite extension \(L/K\) is separable if and only if \(L=KL^p\), i.e. any \(\alpha\in L\) can be written as \(\alpha=c\beta^p\) where \(c\in K\) and \(\beta\in L\).)


Exercise 3.9 Which of the following field extensions \(L/K\) are normal? Justify your answers.

\(\;(a)\;\) \(L=\mathbb{Q}(\sqrt[4]{2})\) over \(K=\mathbb{Q},\hspace{5em}\) \(\;(b)\;\) \(L=K(\sqrt[6]{2})\) over \(K=\mathbb{Q}(\sqrt{-3}),\)

\(\;(c)\;\) \(L=K(\sqrt[4]{2},\sqrt[4]{3})\) over \(K=\mathbb{Q}(i),\hspace{2.05em}\) \(\;(d)\;\) \(L=\mathbb{C}\) over \(K=\mathbb{R},\)

\(\;(e)\;\) \(L=\mathbb{Q}(\sqrt[8]{2},i)\) over \(K=\mathbb{Q},\hspace{4.05em}\) \(\;(f)\;\) \(L=\mathbb{Q}(\sqrt[14]{2})\) over \(K=\mathbb{Q}(\sqrt[7]{2}),\)

\(\;(g)\;\) \(L=\mathbb{Q}(\sqrt[4]{2},i)\) over \(K=\mathbb{Q},\hspace{4.05em}\) \(\;(h)\;\) \(L=\mathbb{Q}(x)\) over \(K=\mathbb{Q}(x^3),\)

\(\;(i)\;\) \(L=\mathbb{C}(x)\) over \(K=\mathbb{C}(x^5),\hspace{4.15em}\) \(\;(j)\;\) \(L=\mathbb{F}_5(x)\) over \(K=\mathbb{F}_5(x^4).\)


Exercise 3.10 Find a normal closure \(N\) for the following extensions \(L/K\).

\(\;(a)\;\) \(L=\mathbb{Q}(\sqrt[4]{2})\) over \(K=\mathbb{Q},\hspace{4em}\) \(\;(b)\;\) \(L=\mathbb{Q}(\sqrt[6]{2},\sqrt{-3})\) over \(K=\mathbb{Q}(\sqrt{-3}),\)

\(\;(c)\;\) \(L=\mathbb{Q}(e^{2\pi i/5})\) over \(K=\mathbb{Q},\hspace{3.1em}\) \(\;(d)\;\) \(L=\mathbb{Q}(x)\) over \(K=\mathbb{Q}(x^3),\)

\(\;(e)\;\) \(L=\mathbb{Q}(x)\) over \(K=\mathbb{Q}(x^4),\hspace{2.95em}\) \(\;(f)\;\) \(L=\mathbb{F}_3(x)\) over \(K=\mathbb{F}_3(x^4),\)

\(\;(g)\;\) \(L=\mathbb{F}_2(x)\) over \(K=\mathbb{F}_2(x^4).\)


Exercise 3.11 Suppose that \(\alpha, \beta\) are algebraic over a field \(K\) and furthermore, \(K(\alpha)/K\) is normal and \(K(\alpha)\cap K(\beta)=K\). Show that \[[K(\alpha,\beta):K]=[K(\alpha):K)] [K(\beta):K].\]


Exercise 3.12 Suppose that \(K\) is a field of characteristic \(p>0\) and \(a\in K\). Given a root \(\theta\) of the polynomial \(f(x)=x^p-x+a\in K[x]\), show that \(K(\theta)/K\) is normal and separable.


Exercise 3.13 Let \(L=\mathbb{F}_p(s,t)\) be the field of rational expressions in two indeterminates \(s, t\) with coefficients in \(\mathbb{F}_p\) where \(p\) is prime. Consider the subfield \(K=\mathbb{F}_p(s^p,t^p)\) of \(L\).

\(\;(a)\;\) Show that \([L:K]=p^2\).

\(\;(b)\;\) Show that if \(\gamma\in L\), then \(\gamma^p\in K\). Deduce that \(L/K\) is not simple.