Theorem: Let $K$ be an algebraic extension of $k$, contained in an algebraic closure $k^a$ of $k$. Then the following conditions are equivalent:
NOR 1. Every embedding of $K$ in $k^a$ over $k$ induces an automorphism of $K$.
NOR 2. $K$ is the splitting field of a family of polynomials in $k[X]$.
NOR 3. Every irreducible polynomial of $k[X]$ which has root in $K$ splits into linear factors in $K$.
Definition: An extension $K$ of $k$ satisfying the hypotheses NOR 1, NOR 2, NOR 3 will be said to be normal.
This is the excerpt from Lang's "Algebra" book. I understood the proof of the theorem. But I cannot understand some moments from definition of normal extension.
1) More precisely, In the definition of normal extension are there any restrictions on $K$? In order to satisfy conditions 1-3 I guess $K$ should be subfield of $k^a$. Am I right or not? Because without this conditions it seems pointless.
2) Lang says that any extension of degree $2$ is normal. Let $k$ be a field and $k\subset K$ with $[K:k]=2$ then it follows that $K$ is algebraic over $k$. But we don't know is $K$ a subfield of $k^a$. If we know this information then all these conditions are equivalent.
Would be very grateful for detailed explanantion! I hope that I put the question correctly.