Question

If F has characteristic 0 show that K has characteristic 0.

Short answer

It is proved that K has characteristic 0.

Step-by-step solution

Definition of homomorphism 

Homomorphism is a transformation of one set into another that preserves in the second set the relation between element of first

Showing that  K has characteristic 0

Let K is a extension of F . A field homomorphism $\varphi$ defined as:

localid="1659036471411" $\varphi :F\to K$

Under which localid="1659036479606" $0_F\to 0_F$andlocalid="1659036486677" $1_F\to 1_K$

Let localid="1659036490724" $\text{char}K=m$and localid="1659036494719" $\text{char}F=n$

As localid="1659036498526" $\text{char}F=0$this mean .localid="1659036502121" $n=0$

Therefore,

localid="1659036506390" $\begin{array}{rcl}\varphi \left(0_F\right)& =& 0_F\\ & =& m·1_F\\ & =& m·\varphi \left(1_F\right)\\ & =& \varphi \left(m·1_F\right)\\ & & \end{array}$

And infectivity of localid="1659036509535" $\varphi$implies:

localid="1659036514345" $\varphi \left(0_F\right)=\left(m·1_F\right)$

Which is a contradiction. Therefore localid="1659036520773" $\text{char}K=0$

It is proved that K has characteristic 0.