Exercise 1, § 18

Let D be a field.

a) Let E be a proper subfield of D whose multiplicative group $E^*$ has finite index in $D^*$. Prove that D is finite (for a sequence $(a_n)$ of distinct elements of E and an element $x$ of D - E, consider the classes mod $E^*$ of the elements $x+a_n$).

b) Prove that an element of D that has only finitely many conjugates belongs to the center Z of D (apply a) to the commutant of this element in D, which is a field). c) Deduce that a polynomial in Z[X] that has a root in D - Z has infinitely many roots in D - Z.