Let F lessthanorequalto K be field extension and let a1 a2 e

Let F lessthanorequalto K be field extension and let a_1, a_2, ellipsis, a_n in K be algebraic over F. Show that F[a_1, a_2, ellipsis, a_n] is a subfield of K and F lessthanorequalto F[a_1, ellipsis, a_n] is finite.

Solution

Since a_i is algebraic over F,

it is algebraic over F(a₁, a₂, . . . , a_i1).

Thus [F(a₁, a₂, . . . , a_i) : F(a₁, a₂, . . . , a_i1)] is finite for all i. Therefore the field F(a₁, a₂, . . . , a_n) is a finite extension of F. Hence it is algebraic.

Also if a, b K be algebraic over F. Then F(a, b) is a finite extension of F. Hence all elements of F(a, b) are algebraic over F. In particular, a ± b, ab and a/b if b 6= 0, are all algebraic over F.