Editing Normal extension

{{Short description|Algebraic field extension}}
{{Other uses|Normal closure (disambiguation){{!}}Normal closure}}
In [[abstract algebra]], a '''normal extension''' is an [[Algebraic extension|algebraic field extension]] ''L''/''K'' for which every [[irreducible polynomial]] over ''K'' that has a [[zero of a function|root]] in ''L'' splits into linear factors over ''L''.{{sfn|Lang|2002|p=237|loc=Theorem 3.3, NOR 3}}{{sfn|Jacobson|1989|p=489|loc=Section 8.7}} This is one of the conditions for an algebraic extension to be a [[Galois extension]]. [[Nicolas Bourbaki|Bourbaki]] calls such an extension a '''quasi-Galois extension'''. For [[Finite extension|finite extensions]], a normal extension is identical to a [[splitting field]].

== Definition ==
Let ''<math>L/K</math>'' be an algebraic extension (i.e., ''L'' is an algebraic extension of ''K''), such that <math>L\subseteq \overline{K}</math> (i.e., ''L'' is contained in an [[algebraic closure]] of ''K''). Then the following conditions, any of which can be regarded as a definition of '''normal extension''', are equivalent:{{sfn|Lang|2002|p=237|loc=Theorem 3.3}}
* Every [[Embedding (field theory)|embedding]] of ''L'' in <math>\overline{K}</math> over ''K'' induces an [[automorphism]] of ''L''.
* ''L'' is the [[splitting field]] of a family of polynomials in <math>K[X]</math>.
* Every irreducible polynomial of <math>K[X]</math> that has a root in ''L'' splits into linear factors in ''L''.

== Other properties ==

Let ''L'' be an extension of a field ''K''. Then:

* If ''L'' is a normal extension of ''K'' and if ''E'' is an intermediate extension (that is, ''L''&nbsp;⊇&nbsp;''E''&nbsp;⊇&nbsp;''K''), then ''L'' is a normal extension of ''E''.{{sfn|Lang|2002|p=238|loc=Theorem 3.4}}
* If ''E'' and ''F'' are normal extensions of ''K'' contained in ''L'', then the [[compositum]] ''EF'' and ''E''&nbsp;∩&nbsp;''F'' are also normal extensions of ''K''.{{sfn|Lang|2002|p=238|loc=Theorem 3.4}}

== Equivalent conditions for normality ==

Let <math>L/K</math> be algebraic. The field ''L'' is a normal extension if and only if any of the equivalent conditions below hold.

* The [[Minimal polynomial (field theory)|minimal polynomial]] over ''K'' of every element in ''L'' splits in ''L'';
* There is a set <math>S \subseteq K[x]</math> of polynomials that each splits over ''L'', such that if <math>K\subseteq F\subsetneq L</math> are fields, then ''S'' has a polynomial that does not split in ''F'';
* All homomorphisms <math>L \to \bar{K}</math> that fix all elements of ''K'' have the same image;
* The group of automorphisms, <math>\text{Aut}(L/K),</math> of ''L'' that fix all elements of ''K'', acts transitively on the set of homomorphisms <math>L \to \bar{K}</math> that fix all elements of ''K''.

== Examples and counterexamples ==

For example, <math>\Q(\sqrt{2})</math> is a normal extension of <math>\Q,</math> since it is a splitting field of <math>x^2-2.</math> On the other hand, <math>\Q(\sqrt[3]{2})</math> is not a normal extension of <math>\Q</math> since the irreducible polynomial <math>x^3-2</math> has one root in it (namely, <math>\sqrt[3]{2}</math>), but not all of them (it does not have the non-real cubic roots of&nbsp;2). Recall that the field <math>\overline{\Q}</math> of [[algebraic number]]s is the algebraic closure of <math>\Q,</math> and thus it contains <math>\Q(\sqrt[3]{2}).</math> Let <math>\omega</math> be a primitive cubic root of unity. Then since, 
<math display="block">\Q (\sqrt[3]{2})=\left. \left \{a+b\sqrt[3]{2}+c\sqrt[3]{4}\in\overline{\Q }\,\,\right | \,\,a,b,c\in\Q \right \}</math> 
the map
<math display="block">\begin{cases} \sigma:\Q (\sqrt[3]{2})\longrightarrow\overline{\Q}\\ a+b\sqrt[3]{2}+c\sqrt[3]{4}\longmapsto a+b\omega\sqrt[3]{2}+c\omega^2\sqrt[3]{4}\end{cases}</math>
is an embedding of <math>\Q(\sqrt[3]{2})</math> in <math>\overline{\Q}</math> whose restriction to <math>\Q </math> is the identity. However, <math>\sigma</math> is not an automorphism of <math>\Q (\sqrt[3]{2}).</math>

For any prime <math>p,</math> the extension <math>\Q (\sqrt[p]{2}, \zeta_p)</math> is normal of degree <math>p(p-1).</math> It is a splitting field of <math>x^p - 2.</math> Here <math>\zeta_p</math> denotes any <math>p</math>th [[primitive root of unity]]. The field <math>\Q (\sqrt[3]{2}, \zeta_3)</math> is the normal closure (see below) of <math>\Q (\sqrt[3]{2}).</math>

==Normal closure==

If ''K'' is a field and ''L'' is an algebraic extension of ''K'', then there is some algebraic extension ''M'' of ''L'' such that ''M'' is a normal extension of ''K''. Furthermore, [[up to isomorphism]] there is only one such extension that is minimal, that is, the only subfield of ''M'' that contains ''L'' and that is a normal extension of ''K'' is ''M'' itself. This extension is called the '''normal closure''' of the extension ''L'' of ''K''.

If ''L'' is a [[finite extension]] of ''K'', then its normal closure is also a finite extension.

== See also ==

* [[Galois extension]]
* [[Normal basis]]

== Citations ==
{{reflist}}

== References ==
* {{Lang Algebra|3rd}}
* {{citation | last = Jacobson | first = Nathan | author-link = Nathan Jacobson | title = Basic Algebra II| edition = 2nd | year = 1989 | publisher = W.&nbsp;H. Freeman | isbn = 0-7167-1933-9 | mr = 1009787}}

[[Category:Field extensions]]