Abstract Algebra: Theory and Applications · Problem 18.4.10

Lattices and Boolean Algebras — Thomas Judson

Mathematics — Abstract Algebra · prime-subfield

Problem

A field

F

is called a prime field if it has no proper subfields. If

E

is a subfield of

F

and

E

is a prime field, then

E

is a prime subfield of

F

. ((a)) Prove that every field contains a unique prime subfield. ((b)) If

F

is a field of characteristic

0

, prove that the prime subfield of

F

is isomorphic to the field of rational numbers,

\mathbb{Q}

. ((c)) If

F

is a field of characteristic

p

, prove that the prime subfield of

F

is isomorphic to

\mathbb{Z}_p