Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Fall 2006 • Question 9

Question

Math 6310 • Algebra Fall 2006 • Question 9

Let \(V\) be a finite-dimensional vector space over a field of characteristic \(p>0\). Let \(T:V\longrightarrow V\) be a linear transformation with \(T^{p^n}=\operatorname {Id}\) for some \(n\). Prove that there is a nonzero vector \(v\in V\) with \(Tv=v\).

Raw LaTeX

Let \(V\) be a finite-dimensional vector space over a field of characteristic \(p>0\).
Let \(T:V\longrightarrow V\) be a linear transformation with \(T^{p^n}=\operatorname{Id}\) for
some \(n\). Prove that there is a nonzero vector \(v\in V\) with \(Tv=v\).

Topic Tags

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