Tutorials for Algebra-III

Show that \(p(x)=x^3-2x-2\) is irreducible in \(\Q[x]\text{.}\) If \(\theta\) is a root of this polynomial, compute \((1+\theta)(1+\theta+\theta^2)\text{,}\) and \((1+\theta)(1+\theta+\theta^2)^{-1}\) in \(\Q(\theta)\text{.}\)

Show that \(x^3+x+1\) is irreducible over \(\mathbb{F}_2\text{,}\) and let \(\theta\) be its root. Compute the powers of \(\theta\) in \(\mathbb{F}_2(\theta)\text{.}\) Find the degree of the field extension \(\mathbb{F}_2(\theta)/\mathbb{F}_2\text{.}\)

Show that if \(a\in\Q\) is a root of a monic polynomial in \(\Z[x]\text{,}\) then \(a\in\Z\text{.}\)

Show that \(x^5-ax-1\in\Z[x]\) is irreducible for all \(a\in\Z\setminus\{0,-1,2\}\text{.}\)