In Chapter 5.3 of Lang's Algebra, there is a theorem claims that these three conditions of normal extensions are equivalent:
My question is about how we can deduce NOR 1 assuming NOR 2. It seemed to me at first that this fact could be proved using Corollary 3.2 of the same chapter:
I guess there is something wrong in my proof because Lang himself did not give a proof in this way. But I can't figured out what exactly is wrong in my proof. The following is my proof:
The algebraic closure $k^a$ of $k$ is also an algebraic closure $K^a$ of $K$, since $k^a$ is naturally an algebraic closed field and we can deduce that $k^a$ is algebraic over $K$ by the fact that $k[X]$ is a subring of $K[X]$. Then we use Corollary 3.2 of the same chapter.
In Corollary 3.2 we let $E=K$. Thus every embedding of $K$ into $k^a$ (i.e. $K^a$) over $k$ induces an automorphism of $K$ as we wish.
Could anyone show me where the error is, or (highly impossible as I believe) there is nothing wrong about my proof? Thanks in advance!
edit:
The following is Lang's proof of NOR 1 assuming NOR 2, where one can see he avoided to use Corollary 3.2.
