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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2012 • Question 5

Question

Math 6310 • Algebra Fall 2012 • Question 5

Let \(\phi \) denote the Frobenius map \(x\mapsto x^p\) on the finite field \(\mathbb {F}_{p^n}\). Determine the Jordan canonical form (over \(\overline {\mathbb {F}}_p\)) for \(\phi \) regarded as an \(\mathbb {F}_p\)-linear transformation of \(\mathbb {F}_{p^n}\).

Raw LaTeX

Let \(\phi\) denote the Frobenius map \(x\mapsto x^p\) on the finite field \(\mathbb{F}_{p^n}\).
Determine the Jordan canonical form (over \(\overline{\mathbb{F}}_p\)) for \(\phi\) regarded as an
\(\mathbb{F}_p\)-linear transformation of \(\mathbb{F}_{p^n}\).

Topic Tags

Finite Fields Matrix Similarity, Jordan Forms, and Finite-Order Diagonalization