Question

Prove that every field of characteristic 0 is infinite.

Short answer

Every field with characteristic 0 is finite.

Step-by-step solution

Definition of ring

A field is a commutative ring with unity in which every nonzero element has a multiplicative inverse.

Showing that every field with characteristic 0 is finite

Let F be a field and $charF=0$

A field is said to heve characteristic zero if $n{1}_F\ne 0$ for every positive n . So let the characteristic of field F is n.

Then charF=0 therefore n=0 .

Also as a field is a commutative ring with unity then any two nonzero element of F have same additive order. That is if $a,b\left(\ne 0\right)\in F$.

Then $\left|a\right|=\left|b\right|$.

In particular$\left|a\right|=1$for every$\left(0\ne a\right)\in F$.

Since n=0 therefore order of 1 is infinite and so order of every nonzero element is of infinite order.

Now as every element of field F is infinite therefore the field F is infinite.