8 Problem Sheet 8:
Galois groups of polynomials
Exercise 8.1 Compute the Galois groups of the following cubic polynomials over \(\mathbb{Q}\).
\(\;(a)\;\) \(f(x)=x^3-3x+2,\hspace{4em}\) \(\;(b)\;\) \(f(x)=x^3-39x+26,\)
\(\;(c)\;\) \(f(x)=x^3-117x+53,\hspace{2.55em}\) \(\;(d)\;\) \(f(x)=x^3-84x+56,\)
\(\;(e)\;\) \(f(x)=x^3-63x+9,\hspace{3.45em}\) \(\;(f)\;\) \(f(x)=x^3+6x^2+9x+3.\)
Exercise 8.2 Find all complex roots of the polynomials
\(\;(a)\;\) \(x^4+(1/2)x^2-2x+17/16,\hspace{3em}\) \(\;(b)\;\) \(x^4-(1/2)x^2-\sqrt{15}x+69/16.\)
Exercise 8.3 Find the Galois groups of the following quartic polynomials over \(\mathbb{Q}\).
\(\;(a)\;\) \(f(x)=x^4+2,\hspace{7.7em}\) \(\;(b)\;\) \(f(x)=x^4+4,\)
\(\;(c)\;\) \(f(x)=x^4-x+1,\hspace{5.95em}\) \(\;(d)\;\) \(f(x)=x^4+3x+3,\)
\(\;(e)\;\) \(f(x)=x^4+5x+5,\hspace{5.4em}\) \(\;(f)\;\) \(f(x)=x^4+8x+12,\)
\(\;(g)\;\) \(f(x)=x^4+4x^2-5,\hspace{5.05em}\) \(\;(h)\;\) \(f(x)=x^4+5x^2+5,\)
\(\;(i)\;\) \(f(x)=x^4-5x^2+6,\hspace{5.2em}\) \(\;(j)\;\) \(f(x)=x^4-6x^2+2x+2,\)
\(\;(k)\;\) \(f(x)=x^4+x^3+x^2+x+1,\hspace{1.6em}\) \(\;(l)\;\) \(f(x)=x^4+4x^3+5x^2+2x+1.\)
Exercise 8.4 Let \(f(x)=x^4+ax^2+b\in K[x]\) be an irreducible polynomial over a field \(K\) of characteristic not \(2\) and let \(d=a^2-4b\). Use Theorem 8.5 to show that the Galois group of \(f(x)\) is \[G_f\cong\begin{cases} \mathbb{Z}/2\times\mathbb{Z}/2 & \text{if $b\in {K^\times}^2$}, \\ \mathbb{Z}/4 & \text{if $b\not\in {K^\times}^2\;$ and $\;bd\in {K^\times}^2$}, \\ D_4 & \text{if $b\not\in {K^\times}^2\;$ and $\;bd\not\in {K^\times}^2$}. \end{cases}\]
Exercise 8.5 Let \(K=\mathbb{F}_p(u)\) be the field of rational expressions with indeterminate \(u\) and coefficients in \(\mathbb{F}_p\) for some odd prime \(p\). Find the Galois group of the polynomial \(f(x)=x^4+ux+u\in K[x]\) over \(K\).
Exercise 8.6 Find the Galois groups of the normal closures of the following extensions:
\(\;(a)\;\) \(\mathbb{Q}\left(\sqrt{-5+2\sqrt{5}}\right)/\mathbb{Q},\hspace{3em}\) \(\;(b)\;\) \(\mathbb{Q}\left(\sqrt{1+\sqrt{5}}\right)/\mathbb{Q}.\)
Exercise 8.7 Let \(L\) be a splitting field of \(f(x)=x^4-4x^2+11\) over \(\mathbb{Q}\).
\(\;(a)\;\) Describe the structure of \(G=\Gal(L/\mathbb{Q})\).
\(\;(b)\;\) Find all subfields \(M\subset L\) with \([L:M]=2\) and \(M/\mathbb{Q}\) Galois.
\(\;(c)\;\) Find all subfields \(K\subset L\) with \([K:\mathbb{Q}]=2\). For each such \(K\), describe the structure of \(\Gal(L/K)\).
Exercise 8.8 Let \(L=\mathbb{Q}(\theta_1,\theta_2)\) where \(\theta_1,\theta_2=\sqrt{17\pm 2\sqrt{30}}\).
\(\;(a)\;\) Prove that \(L/\mathbb{Q}\) is a Galois extension.
\(\;(b)\;\) Find \(\Gal(L/\mathbb{Q})\) and describe how its elements act on \(\theta_1,\theta_2\).
\(\;(c)\;\) Find all subfields \(K\subset L\) with \([K:\mathbb{Q}]=2\).
\(\;(d)\;\) Use your result to express \(\theta_1,\theta_2\) in the form \(\sqrt{a}\pm\sqrt{b}\) for integers \(a,b\).
Exercise 8.9 \(\;(a)\;\) Find the minimal Galois extension \(L/\mathbb{Q}\) containing \(\theta=\sqrt{13-5\sqrt{6}}\).
\(\;(b)\;\) Determine the structure of \(\Gal(L/\mathbb{Q})\) and find all conjugates of \(\theta\) over \(\mathbb{Q}\).
\(\;(c)\;\) Find all subfields \(K\subset L\) with \([K:\mathbb{Q}]=2\).
Exercise 8.10 Let \(L\) be a splitting field of \(x^4+2x^2-4\) over \(\mathbb{Q}\).
\(\;(a)\;\) Compute the Galois group \(\Gal(L/\mathbb{Q})\).
\(\;(b)\;\) Show that \(\mathbb{Q}(i,\sqrt{5})\subset L\) and determine \(\Gal(L/\mathbb{Q}(\sqrt{5}))\), \(\Gal(L/\mathbb{Q}(i\sqrt{5}))\) and \(\Gal(L/\mathbb{Q}(i))\).
Exercise 8.11 \(\;(a)\;\) Let \(f(x)\) be an irreducible polynomial in \(\mathbb{Q}[x]\) of prime degree \(p\geq 5\) with exactly two non-real roots. Show that the Galois group of \(f(x)\) is isomorphic to \(S_p\).
\(\;(b)(\star)\;\) Set \(r=p-2\), let \(m\geq 2\) be a positive even integer and \(a_1<a_2<\cdots<a_r\) be distinct even integers. Consider the degree \(p\) polynomial \[f(x)=(x^2+m)(x-a_1)(x-a_2)\cdots(x-a_r)-2.\] \(\;\;\;(i)\;\) Prove that \(f(x)\) is irreducible over \(\mathbb{Q}\).
\(\;\;(ii)\;\) Prove that \(f(x)\) has at least \(p-2\) real roots and has \(2\) non-real roots for sufficiently large \(m\).
\(\qquad\) (Hint: if the roots are \(x_1,...,x_p\), show that \(\sum x_j^2<0\) when \(m\) is large enough.)
\(\qquad\) As a result, \(G_f\cong S_p\) and \(f(x)\) is not solvable by radicals over \(\mathbb{Q}\).