Exercise 5, § 9

Let A be an algebra over a commutative field K.

a) Prove that an element of the radical of A that is algebraic over K is nilpotent. b) Suppose that K is an infinite field and that Card(K) $>[A : K]$. Prove that the radical of A is a nil ideal (reason as in Lemma 1 of VIII, p. 47).