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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2016 • Question 7

Question

Math 6310 • Algebra Fall 2016 • Question 7

Let \(p\) be a prime number, \(n\) a positive integer, and \(V\) the field \(\mathbb {F}_{p^n}\) viewed as a vector space over \(\mathbb {F}_p\). Prove that the Frobenius map \(\varphi :V\longrightarrow V\), assigning \(v\) to \(v^p\), is diagonalizable over \(\mathbb {F}_p\) if and only if \(n\) divides \(p-1\).

Raw LaTeX

Let \(p\) be a prime number, \(n\) a positive integer, and \(V\) the field \(\mathbb{F}_{p^n}\) viewed
as a vector space over \(\mathbb{F}_p\). Prove that the Frobenius map
\(\varphi:V\longrightarrow V\), assigning \(v\) to \(v^p\), is diagonalizable over \(\mathbb{F}_p\) if
and only if \(n\) divides \(p-1\).

Topic Tags

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