Normal extension

<h2 id="definition">Definition</h2>
Let 
 
 
 
 L
 
 /
 
 K
 
 
 {\displaystyle L/K}
 
 be an algebraic extension (i.e., L is an algebraic extension of K), such that 
 
 
 
 L
 ⊆
 
 
 K
 ¯
 
 
 
 
 {\displaystyle L\subseteq {\overline {K}}}
 
 (i.e., L is contained in an <a href="/facts/Algebraic_closure/yjewG1mM">algebraic closure</a> of K). Then the following conditions, any of which can be regarded as a definition of normal extension, are equivalent:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<ul><li>Every <a href="/facts/Embedding_(field_theory)/HKeVKc42">embedding</a> of L in 
 
 
 
 
 
 K
 ¯
 
 
 
 
 {\displaystyle {\overline {K}}}
 
 over K induces an <a href="/facts/Automorphism/WnyxNFkQ">automorphism</a> of L.</li>
<li>L is the <a href="/facts/Splitting_field/m38eWUH1">splitting field</a> of a family of polynomials in 
 
 
 
 K
 [
 X
 ]
 
 
 {\displaystyle K[X]}
 
.</li>
<li>Every irreducible polynomial of 
 
 
 
 K
 [
 X
 ]
 
 
 {\displaystyle K[X]}
 
 that has a root in L splits into linear factors in L.</li></ul>
<h2 id="other-properties">Other properties</h2>
Let L be an extension of a field K. Then:

<ul><li>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.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a></li>
<li>If E and F are normal extensions of K contained in L, then the <a href="/facts/Compositum/FklJQo5j">compositum</a> EF and E ∩ F are also normal extensions of K.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li></ul>
<h2 id="equivalent-conditions-for-normality">Equivalent conditions for normality</h2>
Let 
 
 
 
 L
 
 /
 
 K
 
 
 {\displaystyle L/K}
 
 be algebraic. The field L is a normal extension if and only if any of the equivalent conditions below hold.

<ul><li>The <a href="/facts/Minimal_polynomial_(field_theory)/dJv9Q0Kh">minimal polynomial</a> over K of every element in L splits in L;</li>
<li>There is a set 
 
 
 
 S
 ⊆
 K
 [
 x
 ]
 
 
 {\displaystyle S\subseteq K[x]}
 
 of polynomials that each splits over L, such that if 
 
 
 
 K
 ⊆
 F
 ⊊
 L
 
 
 {\displaystyle K\subseteq F\subsetneq L}
 
 are fields, then S has a polynomial that does not split in F;</li>
<li>All homomorphisms 
 
 
 
 L
 →
 
 
 
 K
 ¯
 
 
 
 
 
 {\displaystyle L\to {\bar {K}}}
 
 that fix all elements of K have the same image;</li>
<li>The group of automorphisms, 
 
 
 
 
 Aut
 
 (
 L
 
 /
 
 K
 )
 ,
 
 
 {\displaystyle {\text{Aut}}(L/K),}
 
 of L that fix all elements of K, acts transitively on the set of homomorphisms 
 
 
 
 L
 →
 
 
 
 K
 ¯
 
 
 
 
 
 {\displaystyle L\to {\bar {K}}}
 
 that fix all elements of K.</li></ul>
<h2 id="examples-and-counterexamples">Examples and counterexamples</h2>
For example, 
 
 
 
 
 Q
 
 (
 
 
 2
 
 
 )
 
 
 {\displaystyle \mathbb {Q} ({\sqrt {2}})}
 
 is a normal extension of 
 
 
 
 
 Q
 
 ,
 
 
 {\displaystyle \mathbb {Q} ,}
 
 since it is a splitting field of 
 
 
 
 
 x
 
 2
 
 
 −
 2.
 
 
 {\displaystyle x^{2}-2.}
 
 On the other hand, 
 
 
 
 
 Q
 
 (
 
 
 2
 
 3
 
 
 
 )
 
 
 {\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}})}
 
 is not a normal extension of 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
 since the irreducible polynomial 
 
 
 
 
 x
 
 3
 
 
 −
 2
 
 
 {\displaystyle x^{3}-2}
 
 has one root in it (namely, 
 
 
 
 
 
 2
 
 3
 
 
 
 
 
 {\displaystyle {\sqrt[{3}]{2}}}
 
), but not all of them (it does not have the non-real cubic roots of 2). Recall that the field 
 
 
 
 
 
 
 Q
 
 ¯
 
 
 
 
 {\displaystyle {\overline {\mathbb {Q} }}}
 
 of <a href="/facts/Algebraic_number/MkH1PzTK">algebraic numbers</a> is the algebraic closure of 
 
 
 
 
 Q
 
 ,
 
 
 {\displaystyle \mathbb {Q} ,}
 
 and thus it contains 
 
 
 
 
 Q
 
 (
 
 
 2
 
 3
 
 
 
 )
 .
 
 
 {\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}}).}
 
 Let 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 be a primitive cubic root of unity. Then since,

Q
        
        (
        
          
            2
            
              3
            
          
        
        )
        =
        
          
          
            
              {
              
                a
                +
                b
                
                  
                    2
                    
                      3
                    
                  
                
                +
                c
                
                  
                    4
                    
                      3
                    
                  
                
                ∈
                
                  
                    
                      Q
                    
                    ¯
                  
                
                
                
              
              |
            
            
            
            a
            ,
            b
            ,
            c
            ∈
            
              Q
            
          
          }
        
      
    
    {\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}})=\left.\left\{a+b{\sqrt[{3}]{2}}+c{\sqrt[{3}]{4}}\in {\overline {\mathbb {Q} }}\,\,\right|\,\,a,b,c\in \mathbb {Q} \right\}}
  
 
the map

{
            
              
                
                  σ
                  :
                  
                    Q
                  
                  (
                  
                    
                      2
                      
                        3
                      
                    
                  
                  )
                  ⟶
                  
                    
                      
                        Q
                      
                      ¯
                    
                  
                
              
              
                
                  a
                  +
                  b
                  
                    
                      2
                      
                        3
                      
                    
                  
                  +
                  c
                  
                    
                      4
                      
                        3
                      
                    
                  
                  ⟼
                  a
                  +
                  b
                  ω
                  
                    
                      2
                      
                        3
                      
                    
                  
                  +
                  c
                  
                    ω
                    
                      2
                    
                  
                  
                    
                      4
                      
                        3
                      
                    
                  
                
              
            
            
          
        
      
    
    {\displaystyle {\begin{cases}\sigma :\mathbb {Q} ({\sqrt[{3}]{2}})\longrightarrow {\overline {\mathbb {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}}}

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

For any prime 
 
 
 
 p
 ,
 
 
 {\displaystyle p,}
 
 the extension 
 
 
 
 
 Q
 
 (
 
 
 2
 
 p
 
 
 
 ,
 
 ζ
 
 p
 
 
 )
 
 
 {\displaystyle \mathbb {Q} ({\sqrt[{p}]{2}},\zeta _{p})}
 
 is normal of degree 
 
 
 
 p
 (
 p
 −
 1
 )
 .
 
 
 {\displaystyle p(p-1).}
 
 It is a splitting field of 
 
 
 
 
 x
 
 p
 
 
 −
 2.
 
 
 {\displaystyle x^{p}-2.}
 
 Here 
 
 
 
 
 ζ
 
 p
 
 
 
 
 {\displaystyle \zeta _{p}}
 
 denotes any 
 
 
 
 p
 
 
 {\displaystyle p}
 
th <a href="/facts/Primitive_root_of_unity/EqH0ho1Y">primitive root of unity</a>. The field 
 
 
 
 
 Q
 
 (
 
 
 2
 
 3
 
 
 
 ,
 
 ζ
 
 3
 
 
 )
 
 
 {\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}},\zeta _{3})}
 
 is the normal closure (see below) of 
 
 
 
 
 Q
 
 (
 
 
 2
 
 3
 
 
 
 )
 .
 
 
 {\displaystyle \mathbb {Q} ({\sqrt[{3}]{2}}).}

<h2 id="normal-closure">Normal closure</h2>
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, <a href="/facts/Up_to_isomorphism/ydz4ztzX">up to isomorphism</a> 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 <a href="/facts/Finite_extension/hY5LgHYk">finite extension</a> of K, then its normal closure is also a finite extension.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Galois_extension/52fzNQbv">Galois extension</a></li>
<li><a href="/facts/Normal_basis/3bhymK7j">Normal basis</a></li></ul>
<h2 id="citations">Citations</h2>

<ul><li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (2002), Algebra, <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>, vol. 211 (Revised third ed.), New York: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-95385-4, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1878556">1878556</a></li>
<li><a href="/facts/Nathan_Jacobson/TEDrg4Lk">Jacobson, Nathan</a> (1989), Basic Algebra II (2nd ed.), W. H. Freeman, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7167-1933-9, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1009787">1009787</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Lang 2002, p. 237, Theorem 3.3, NOR 3. - Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1878556" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1878556</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Jacobson 1989, p. 489, Section 8.7. - Jacobson, Nathan (1989), Basic Algebra II (2nd ed.), W. H. Freeman, ISBN 0-7167-1933-9, MR 1009787 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1009787" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1009787</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Lang 2002, p. 237, Theorem 3.3. - Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1878556" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1878556</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Lang 2002, p. 238, Theorem 3.4. - Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1878556" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1878556</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Lang 2002, p. 238, Theorem 3.4. - Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1878556" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1878556</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
</ol>

Normal extension open-in-new

Normal extension