1 Problem Sheet 1:
Fields and polynomials
Exercise 1.1 \(\;(a)\;\) Make a list of all (monic) irreducible polynomials of degrees at most \(4\) in \(\mathbb{F}_2[x]\), i.e. polynomials over the field \(\mathbb{F}_2\) of \(2\) elements.
\(\;(b)\;\) Using part \((a)\) and without finding them directly, how many irreducible polynomials of degree \(5\) are there in \(\mathbb{F}_2[x]\)?
\(\;(c)\;\) Find the polynomials from part \((b)\).
Exercise 1.2 \(\;\)Decompose the following into a product of irreducible factors in the given polynomial ring:
\(\;(a)\;\;x^3+2x+3\) in \(\mathbb{F}_5[x],\hspace{2em}\) \(\;(b)\;\;2x^3+x^2+2x+2\) in \(\mathbb{F}_5[x],\)
\(\;(c)\;\;x^3+2x+3\) in \(\mathbb{F}_7[x],\hspace{2em}\) \(\;(d)\;\;2x^3+x^2+2x+2\) in \(\mathbb{F}_7[x],\)
\(\;(e)\;\;x^4+64\) in \(\mathbb{Q}[x],\hspace{4em}\) \(\;(f)\;\;x^3+2\) in \(\mathbb{Q}[x],\)
\(\;(g)\;\;x^4+1\) in \(\mathbb{R}[x],\hspace{4.5em}\) \(\;(h)\;\;x^6+27\) in \(\mathbb{R}[x],\)
\(\;(i)\;\;x^7+1\) in \(\mathbb{F}_2[x],\hspace{4.35em}\) \(\;(j)\;\;x^3+2\) in \(\mathbb{F}_3[x],\)
\(\;(k)\;\;x^4+2\) in \(\mathbb{F}_3[x],\hspace{4.25em}\) \(\;(l)\;\;x^4+2\) in \(\mathbb{F}_{13}[x].\)
Exercise 1.3 \(\;\)Find all prime numbers \(p\) for which \(x+2\) is a factor of \(x^4+x^3+x^2-x+1\) in \(\mathbb{F}_p[x]\).
Exercise 1.4 \(\;\)How many irreducible monic quadratic polynomials are there in \(\mathbb{F}_p[x]\)?
Exercise 1.5 \((\star)\;\) Suppose we are given \(n\geq 1\) distinct integers \(a_1,...,a_n\in\mathbb{Z}\). Show that the polynomial \[f(x)=(x-a_1)(x-a_2)...(x-a_n)-1\in\mathbb{Q}[x]\] is irreducible in \(\mathbb{Q}[x]\). (Hint: use Gauss’s Lemma and construct a polynomial with too many roots).
Exercise 1.6 \(\;\)Find generators for each of the following ideals in the given polynomial rings:
\(\;(a)\;\) \(I=\left\{f(x)\in\mathbb{Q}[x] \;|\; f(i)=0 \right\}\subset \mathbb{Q}[x],\)
\(\;(b)\;\) \(I=\left\{f(x)\in\mathbb{Q}[x] \;|\; f(i\sqrt{2})=0 \right\}\subset \mathbb{Q}[x],\)
\(\;(c)\;\) \(I=\left\{f(x)\in\mathbb{Q}[x] \;|\; f(\sqrt{2})=f(\sqrt{5})=0 \right\}\subset \mathbb{Q}[x],\)
\(\;(d)\;\) \(I=\left\{f(x)\in\mathbb{R}[x] \;|\; f(\sqrt{2})=0 \right\}\subset \mathbb{R}[x],\)
\(\;(e)\;\) \(I=\left\{f(x)\in\mathbb{R}[x] \;|\; f(i\sqrt{2})=0 \right\}\subset \mathbb{R}[x],\)
\(\;(f)\;\) \(I=\left\{f(x)\in\mathbb{R}[x] \;|\; f(i)=f(\sqrt{2})=f(1)=0 \right\}\subset \mathbb{R}[x].\)
Exercise 1.7 \(\;\)Use Gauss’s Lemma to show that the following polynomials are irreducible in \(\mathbb{Q}[x]\):
\(\;(a)\;\;x^4+8,\hspace{4em}\) \(\;(b)\;\;x^4-5x^2+2,\hspace{4em}\) \(\;(c)\;\;x^4-22x^2+1.\)
Exercise 1.8 \(\;\)Which of the following subsets of \(\mathbb{C}\) are fields with respect to the usual addition and multiplication:
\(\;(a)\;\;\mathbb{Z},\hspace{4em}\) \(\;(b)\;\;\{0,\pm 1\},\hspace{4em}\) \(\;(c)\;\;\{0\},\)
\(\;(d)\;\;\left\{a+b\sqrt{2} \;|\; a, b\in\mathbb{Q}\right\},\hspace{3em}\) \(\;(e)\;\;\left\{a+b\sqrt[3]{2} \;|\; a, b\in\mathbb{Q}\right\},\)
\(\;(f)\;\;\left\{a+b\sqrt[4]{2} \;|\; a, b\in\mathbb{Q}\right\},\hspace{3em}\) \(\;(g)\;\;\left\{a+b\sqrt{2} \;|\; a, b\in\mathbb{Z}\right\},\)
\(\;(h)\;\;\{z\in\mathbb{C} \;|\; |z|\leq 1\}.\)
Exercise 1.9 \(\;\)Show that every subfield of \(\mathbb{C}\) contains the rational numbers \(\mathbb{Q}\).
Exercise 1.10 \(\;\)Give an example of an infinite field of non-zero characteristic \(p\).
Exercise 1.11 \(\;(a)\;\) Let \(K\subset L\) be a field extension and suppose \(\theta\in L\) and \(\theta^2\in K\) but \(\theta\not\in K\). Show that \[K(\theta)=\left\{a+b\theta \;|\; a, b \in K\right\}\] is the minimal subfield of \(L\) containing \(K\) and \(\theta\).
\(\;(b)\;\) Suppose \(r, s\neq 0\) are two non-zero elements in a field \(K\). Show that \(K(\sqrt{r})=K(\sqrt{s})\) if and only if \(rs\) is a square in \(L\), i.e. \(rs=t^2\) for some \(t\in K\).
Exercise 1.12 \(\;\)Suppose \(K\) is a field of characteristic \(p>0\).
\(\;(a)\;\) Write the polynomial \(x^p-x\in K[x]\) as a product of irreducible factors.
\(\;(b)\;\) Given an arbitrary element \(a\in K\), show that the polynomial \(x^p-x+a\in K[x]\) has either no roots in \(K\) or \(p\) distinct roots in \(K\).
Exercise 1.13 \((\star)\;\) Let \(p\) be a prime and \(n\geq 2\). Are the following polynomials irreducible in \(\mathbb{Q}[x]\)? Give reasons. \(\;\;(a)\;\;x^n+p^2x+p,\hspace{2cm}\) \(\;\;(b)\;\;x^n+px+p^2.\)
Exercise 1.14 \((\star)\;\) Consider the polynomial \(f(x)=x^4+1\).
\((a)\;\) Show that \(f(x)\) is irreducible in \(\mathbb{Q}[x]\).
\((b)\;\) Show that \(f(x)\) is reducible in \(\mathbb{F}_p[x]\) for every prime \(p\) by expressing it as the product of two quadratic polynomials. (You may use the fact that for every \(p\), at least one of \(\{-1,2,-2\}\) is a square in \(\mathbb{F}_p\).)