Lemma 10.35.10 (00FT)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.35: Jacobson rings
Lemma 10.35.10 (cite)

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Lemma 10.35.10. Suppose that $k$ is a field and suppose that $V$ is a nonzero vector space over $k$. Assume the dimension of $V$ (which is a cardinal number) is smaller than the cardinality of $k$. Then for any linear operator $T : V \to V$ there exists some monic polynomial $P(t) \in k[t]$ such that $P(T)$ is not invertible.

Proof. If not then $V$ inherits the structure of a vector space over the field $k(t)$. But the dimension of $k(t)$ over $k$ is at least the cardinality of $k$ for example due to the fact that the elements $\frac{1}{t - \lambda }$ are $k$-linearly independent. $\square$

Comments (2)

Comment #4 by Johan on July 20, 2012 at 18:10

Spelling: "Asssume".

Comment #16 by Johan on July 22, 2012 at 22:53

Fixed

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