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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2015 • Question 7

Question

Math 6310 • Algebra Fall 2015 • Question 7

Prove that \(\mathbb {F}=\mathbb {Z}[t]/\langle 3, t^3-t^2+1\rangle \) is a field. Find the number of solutions of \(x^{13}+1=0\) in \(\mathbb {F}\), and also the number of solutions of \(x^{13}-1=0\) in \(\mathbb {F}\).

Raw LaTeX

Prove that \(\mathbb{F}=\mathbb{Z}[t]/\langle 3, t^3-t^2+1\rangle\) is a field. Find the number
of solutions of \(x^{13}+1=0\) in \(\mathbb{F}\), and also the number of solutions of
\(x^{13}-1=0\) in \(\mathbb{F}\).

Topic Tags

Finite Fields