Normal extension

Let $L/K$ be an algebraic extension (i.e., L is an algebraic extension of K), such that $L\backslash subseteq\; \backslash overline\{K\}$ (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 of L in $\backslash overline\{K\}$ over K induces an automorphism of L.
• L is the splitting field of a family of polynomials in $K[X]$.
• Every irreducible polynomial of $K[X]$ that has a root in L splits into linear factors in L.

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 ⊇ E ⊇ 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 ∩ F are also normal extensions of K.{{sfn|Lang|2002|p=238|loc=Theorem 3.4}}

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

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

For example, $\backslash Q(\backslash sqrt\{2\})$ is a normal extension of $\backslash Q,$ since it is a splitting field of $x^2-2.$ On the other hand, $\backslash Q(\backslash sqrt[3]\{2\})$ is not a normal extension of $\backslash Q$ since the irreducible polynomial $x^3-2$ has one root in it (namely, $\backslash sqrt[3]\{2\}$), but not all of them (it does not have the non-real cubic roots of 2). Recall that the field $\backslash overline\{\backslash Q\}$ of algebraic numbers is the algebraic closure of $\backslash Q,$ and thus it contains $\backslash Q(\backslash sqrt[3]\{2\}).$ Let $\backslash omega$ be a primitive cubic root of unity. Then since,

$$\backslash Q\; (\backslash sqrt[3]\{2\})=\backslash left.\; \backslash left\; \backslash \{a+b\backslash sqrt[3]\{2\}+c\backslash sqrt[3]\{4\}\backslash in\backslash overline\{\backslash Q\; \}\backslash ,\backslash ,\backslash right\; |\; \backslash ,\backslash ,a,b,c\backslash in\backslash Q\; \backslash right\; \backslash \}$$

the map

$$\backslash begin\{cases\}\; \backslash sigma:\backslash Q\; (\backslash sqrt[3]\{2\})\backslash longrightarrow\backslash overline\{\backslash Q\}\backslash \backslash \; a+b\backslash sqrt[3]\{2\}+c\backslash sqrt[3]\{4\}\backslash longmapsto\; a+b\backslash omega\backslash sqrt[3]\{2\}+c\backslash omega^2\backslash sqrt[3]\{4\}\backslash end\{cases\}$$

is an embedding of $\backslash Q(\backslash sqrt[3]\{2\})$ in $\backslash overline\{\backslash Q\}$ whose restriction to $\backslash Q$ is the identity. However, $\backslash sigma$ is not an automorphism of $\backslash Q\; (\backslash sqrt[3]\{2\}).$

For any prime $p,$ the extension $\backslash Q\; (\backslash sqrt[p]\{2\},\; \backslash zeta\_p)$ is normal of degree $p(p-1).$ It is a splitting field of $x^p\; -\; 2.$ Here $\backslash zeta\_p$ denotes any $p$th primitive root of unity. The field $\backslash Q\; (\backslash sqrt[3]\{2\},\; \backslash zeta\_3)$ is the normal closure (see below) of $\backslash Q\; (\backslash sqrt[3]\{2\}).$

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.

• Galois extension
• Normal basis

{{reflist}}

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

Category:Field extensions