5 Problem Sheet 5:
Finite fields
Exercise 5.1 \(\;(a)\;\) Construct the field \(\mathbb{F}_8\) having \(8\) elements. Find an explicit generator for the multiplicative subgroup \(\mathbb{F}_8^\times\).
\(\;(b)\;\) Repeat the same question for \(\mathbb{F}_9\), the field with \(9\) elements.
\(\;(c)\;\) Repeat the same question for \(\mathbb{F}_{25}\), the field with \(25\) elements.
Exercise 5.2 \(\;(a)\;\) Find all subfields of \(\mathbb{F}_{729}\).
\(\;(b)\;\) Find all subfields of \(\mathbb{F}_{1024}\).
\(\;(c)\;\) Find all subfields of \(\mathbb{F}_{5^{36}}\).
Exercise 5.3 How many distinct roots do the following polynomials have in \(\mathbb{F}_{16}\)?
\(\;(a)\;\) \(f(x)=x^3-1,\hspace{3.3em}\) \(\;(b)\;\) \(f(x)=x^4-1,\)
\(\;(c)\;\) \(f(x)=x^{15}-1,\hspace{3em}\) \(\;(d)\;\) \(f(x)=x^{19}-1.\)
Exercise 5.4 How many distinct roots do the following polynomials have in \(\mathbb{F}_{81}\)?
\(\;(a)\;\) \(f(x)=x^{80}-1,\hspace{2em}\) \(\;(b)\;\) \(f(x)=x^{81}-1,\hspace{2em}\) \(\;(c)\;\) \(f(x)=x^{88}-1,\hspace{2em}\)
Exercise 5.5 Factorise \(f(x)=x^4+1\) into irreducible polynomials over the following finite fields \(K\):
\(\;(a)\;\) \(K=\mathbb{F}_{5},\hspace{3cm}\) \(\;(b)\;\) \(K=\mathbb{F}_{25},\hspace{3cm}\) \(\;(c)\;\) \(K=\mathbb{F}_{125}.\)
Exercise 5.6 How many monic irreducible factors does \(f(x)=x^{255}-1\in\mathbb{F}_2[x]\) have and what are their degrees?
Exercise 5.7 \(\;(a)\;\) Let \(L\) be a splitting field for \(f(x)=x^4+x^3+1\) over \(K=\mathbb{F}_2\). Find \(\Gal(L/K)\).
\(\;(b)\;\) Do the same with \(K=\mathbb{F}_3\).
\(\;(c)\;\) Do the same with \(K=\mathbb{F}_4\).
Exercise 5.8 Let \(K\) be a splitting field of \(f(x)=x^3+2\) over \(\mathbb{F}_{13}\).
\(\;(a)\;\) Given that \(\theta\in K\) is a root of \(f(x)\), show that \(K=\mathbb{F}_{13}(\theta)\).
\(\;(b)\;\) Find the other two roots, expressing them in the form \(a+b\theta+c\theta^2\) where \(a,b,c\in\mathbb{F}_{13}\).
Exercise 5.9 \(\;(a)\;\) Show that the polynomial \(f(x)=x^4+4x^2+2\) is irreducible over \(\mathbb{F}_5\).
\(\;(b)\;\) Given that \(\theta\) is a root of \(f(x)\), explain why all the roots of \(f(x)\) lie in \(\mathbb{F}_5(\theta)\).
\(\;(c)\;\) Find the other three roots, expressing them in the form \(a+b\theta+c\theta^2+d\theta^3\) where \(a,b,c,d\in\mathbb{F}_5\).
Exercise 5.10 Determine whether or not \(\mathbb{F}_{1024}\) contains a primitive \(9\)-th root of unity, i.e. an element \(\alpha\in\mathbb{F}_{1024}\) such that \(\alpha^9=1\) but \(\alpha^3\neq 1\).
Exercise 5.11 Let \(K\) be a finite field. Prove that \(K\) contains a primitive cube root of unity (i.e. an element \(\zeta\neq 1\) such that \(\zeta^3=1\)) if and only if there is a cubic extension \(L/K\) such that \(L=K(\sqrt[3]{\alpha})\) for some \(\alpha\in K\).
Exercise 5.12 Let \(K=\mathbb{F}_{p^n}\) be a finite field, where \(p\) is prime and \(n\geq 1\).
\(\;(a)\;\) Find the sum of all elements of \(K\).
\(\;(b)\;\) Find the product of all non-zero elements of \(K\).
Exercise 5.13 Given that \(m\mid n\), show that \(\mathbb{F}_{p^n}/\mathbb{F}_{p^m}\) is a Galois extension with cyclic Galois group and construct an explicit generator of \(\Gal(\mathbb{F}_{p^n}/\mathbb{F}_{p^m})\).
Exercise 5.14 Let \(K=\mathbb{F}_q\) be a finite field. Prove that there exists at least one irreducible polynomial of each degree \(m\geq 1\) in \(K[x]\).
Exercise 5.15 For each of the following polynomials \(f(x)\in K[x]\), find a splitting field \(L\) for \(f(x)\) over \(K\) and determine the roots of \(f(x)\) in \(L\):
\(\;(a)\;\) \(f(x)=x^2+1\in\mathbb{F}_2[x],\hspace{3cm}\) \(\;(b)\;\) \(f(x)=x^3+x+1\in\mathbb{F}_2[x]\),
\(\;(c)\;\) \(f(x)=x^3-x+2\in\mathbb{F}_3[x]\).
Exercise 5.16 \((a)\;\) Suppose \(K\) is a field of characteristic \(p>0\) and \(a\in K\). Prove that the polynomial \(f(x)=x^p-x-a\) has either \(p\) roots in \(K\) or no roots in \(K\).
\((b)\;\) In the case where \(f(x)=x^p-x-a\) has no roots in \(K\), let \(\theta\) be a root in a splitting field of \(f(x)\) over \(K\). Show that the extension \(K(\theta)/K\) is Galois, describe the elements of its Galois group, and deduce that \(f(x)\) is irreducible over \(K\).
\((c)\;\) Show that \(f(x)=x^p-x+1\) is irreducible over \(\mathbb{F}_p\) and explain why this implies that \(f(x)\) is irreducible over \(\mathbb{Q}\) for each prime \(p\).