Primitive field

Question

I'm returning student and trying to do self-study for abstract algebra. But, because of my weak understanding of the field theory, I'm stuck with this problem.

The problem is the following :

Let $K$ be a primitive field, which is A field $K$ contains no proper subfield, let $L$ and $M$ be extensions of $K$, and let $φ:L→ M$ be a non-zero homomorphism. Show that:

For all $a ∈ K: φ(a) = a$. And, If $p(x) ∈ K[x]$, $b ∈ L$ and $p(b) = 0$, then $p(φ(b)) = 0$

My thought : I know that $φ$ is injective because $φ$ is non-zero homomorphism. But, after this step, I'm stuck with it.

Any hints or explanations would be helpful to me.

Answer 1

Consider the set $I=\{\,x\in K\mid \phi(x)=x\,\}$. Show that $I$ is a subfield, for exmple by showing

• $0\in I$
• $1\in I$
• $a,b\in I\to a+b\in I$
• $a\in I\to -a\in I$
• $a,b\in I\to ab\in I$
• $a\in I, a\ne 0\to a^{-1}\in I$

These points are all routine, maybe the last is slightly tricky. As $K$ is primitive, $I=K$ follows.