Abstract Algebra: Theory and Applications · Problem 21.5.14

Finite Fields — Thomas Judson

Mathematics — Abstract Algebra · algebraic-extensions-transitivity

Problem

Let

K

be an algebraic extension of

E

, and

E

an algebraic extension of

F

. Prove that

K

is algebraic over

F

. [Caution: Do not assume that the extensions are finite.] Hint. Suppose that

E

is algebraic over

F

and

K

is algebraic over

E

. Let

\alpha \in K

. It suffices to show that

\alpha

is algebraic over some finite extension of

F

. Since

\alpha

is algebraic over

E

, it must be the zero of some polynomial

p(x) = \beta_0 + \beta_1 x + \cdots + \beta_n x^n

E[x]

. Hence

\alpha

is algebraic over

F(\beta_0, \ldots, \beta_n)