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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.

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  • $\begingroup$ The theorem specifies $K$ is contained in $k^a$. This answers your first question. As to the second question, any algebraic extension is contained in an algebraic closure ($k^a$ is only defined up to an isomorphism). $\endgroup$ – Bernard Feb 14 at 19:33
  • $\begingroup$ @Bernard, I did not understand at all your comment regarding my first question. Also why any algebraic extensions are in the algebraic closure? It would be great if you can add more details. $\endgroup$ – ZFR Feb 14 at 19:41
  • $\begingroup$ More exactly, any algebraic extension of $k$ is isomorphic to a subextension of an algebraic closure of $k$. Also, I forgot to say there are no restrictions $K$. $\endgroup$ – Bernard Feb 14 at 19:54
  • $\begingroup$ @Bernard, so you mean in the definition of normal extension there is no restriction on $K$, right? For example, $K$ may not be a subfield of $k^a$, right? It really confuses me because if it's not subfield how it can satisfy NOR 1? It cannot be imbedded into $k^a$? $\endgroup$ – ZFR Feb 14 at 19:57
  • $\begingroup$ @Bernard, However, the book of Steven Roman "Field Theory"defines normal extension as an algebraic extension which satisfies one of these properties. This indeed makes sense. $\endgroup$ – ZFR Feb 14 at 20:11

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