Resolvent (Galois theory)

<h2 id="definition">Definition</h2>
Let n be a positive <a href="/facts/Integer/PUwwrawx">integer</a>, which will be the degree of the equation that we will consider, and (X1, ..., Xn) an ordered list of <a href="/facts/Indeterminate_(variable)/R7QAHS4j">indeterminates</a>. 
According to <a href="/facts/Vieta%2527s_formulas/u3eQJWkA">Vieta's formulas</a> this defines the generic monic polynomial of degree n

F
        (
        X
        )
        =
        
          X
          
            n
          
        
        +
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        (
        −
        1
        
          )
          
            i
          
        
        
          E
          
            i
          
        
        
          X
          
            n
            −
            i
          
        
        =
        
          ∏
          
            i
            =
            1
          
          
            n
          
        
        (
        X
        −
        
          X
          
            i
          
        
        )
        ,
      
    
    {\displaystyle F(X)=X^{n}+\sum _{i=1}^{n}(-1)^{i}E_{i}X^{n-i}=\prod _{i=1}^{n}(X-X_{i}),}

where Ei is the i th <a href="/facts/Elementary_symmetric_polynomial/9RFXe9b6">elementary symmetric polynomial</a>.
The <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a> Sn <a href="/facts/Group_action/Vh9VtUUw">acts</a> on the Xi by permuting them, and this induces an action on the polynomials in the Xi. The <a href="/facts/Stabilizer_(group_theory)/Vh9VtUUw">stabilizer</a> of a given polynomial under this action is generally trivial, but some polynomials have a bigger stabilizer. For example, the stabilizer of an elementary symmetric polynomial is the whole <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> Sn. If the stabilizer is non-trivial, the polynomial is fixed by some non-trivial <a href="/facts/Subgroup/r91mBgpa">subgroup</a> G; it is said to be an invariant of G. Conversely, given a subgroup G of Sn, an invariant of G is a resolvent invariant for G if it is not an invariant of any bigger subgroup of Sn.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
Finding invariants for a given subgroup G of Sn is relatively easy; one can sum the <a href="/facts/Orbit_(group_theory)/Vh9VtUUw">orbit</a> of a <a href="/facts/Monomial/6VAPCYXJ">monomial</a> under the action of Sn. However, it may occur that the resulting polynomial is an invariant for a larger group. For example, consider the case of the subgroup G of S4 of order 4, consisting of (12)(34), (13)(24), (14)(23) and the identity (for the notation, see <a href="/facts/Permutation_group/vRsNNwwx">Permutation group</a>). The monomial X1X2 gives the invariant 2(X1X2 + X3X4). It is not a resolvent invariant for G, because being invariant by (12), it is in fact a resolvent invariant for the larger dihedral subgroup D4: ⟨(12), (1324)⟩, and is used to define the <a href="/facts/Resolvent_cubic/cxbIHK9s">resolvent cubic</a> of the <a href="/facts/Quartic_equation/SjjtXkHL">quartic equation</a>.
If P is a resolvent invariant for a group G of <a href="/facts/Index_(group_theory)/ewY2A71Y">index</a> m inside Sn, then its orbit under Sn has order m. Let P1, ..., Pm be the elements of this orbit. Then the polynomial

R
          
            G
          
        
        =
        
          ∏
          
            i
            =
            1
          
          
            m
          
        
        (
        Y
        −
        
          P
          
            i
          
        
        )
      
    
    {\displaystyle R_{G}=\prod _{i=1}^{m}(Y-P_{i})}

is invariant under Sn. Thus, when expanded, its coefficients are polynomials in the Xi that are invariant under the action of the symmetry group and thus may be expressed as polynomials in the elementary symmetric polynomials. In other words, RG is an <a href="/facts/Irreducible_polynomial/PTeuKm4W">irreducible polynomial</a> in Y whose coefficients are polynomial in the coefficients of F. Having the resolvent invariant as a root, it is called a resolvent (sometimes resolvent equation).
Consider now an irreducible polynomial

f
        (
        X
        )
        =
        
          X
          
            n
          
        
        +
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        
          a
          
            i
          
        
        
          X
          
            n
            −
            i
          
        
        =
        
          ∏
          
            i
            =
            1
          
          
            n
          
        
        (
        X
        −
        
          x
          
            i
          
        
        )
        ,
      
    
    {\displaystyle f(X)=X^{n}+\sum _{i=1}^{n}a_{i}X^{n-i}=\prod _{i=1}^{n}(X-x_{i}),}

with coefficients in a given field K (typically the <a href="/facts/Field_of_rationals/VJQmyY05">field of rationals</a>) and roots xi in an <a href="/facts/Algebraically_closed_extension/HuVQClok">algebraically closed field extension</a>. Substituting the Xi by the xi and the coefficients of F by those of f in the above, we get a polynomial 
 
 
 
 
 R
 
 G
 
 
 (
 f
 )
 
 
 (
 Y
 )
 
 
 {\displaystyle R_{G}^{(f)}(Y)}
 
, also called resolvent or specialized resolvent in case of ambiguity). If the <a href="/facts/Galois_group/A0p35p5c">Galois group</a> of f is contained in G, the specialization of the resolvent invariant is invariant by G and is thus a root of 
 
 
 
 
 R
 
 G
 
 
 (
 f
 )
 
 
 (
 Y
 )
 
 
 {\displaystyle R_{G}^{(f)}(Y)}
 
 that belongs to K (is rational on K). Conversely, if 
 
 
 
 
 R
 
 G
 
 
 (
 f
 )
 
 
 (
 Y
 )
 
 
 {\displaystyle R_{G}^{(f)}(Y)}
 
 has a rational root, which is not a multiple root, the Galois group of f is contained in G.

<h2 id="terminology">Terminology</h2>
There are some variants in the terminology.

<ul><li>Depending on the authors or on the context, resolvent may refer to resolvent invariant instead of to resolvent equation.</li>
<li>A Galois resolvent is a resolvent such that the resolvent invariant is linear in the roots.</li>
<li>The Lagrange resolvent may refer to the linear polynomial 
 
 
 
 
 ∑
 
 i
 =
 0
 
 
 n
 −
 1
 
 
 
 X
 
 i
 
 
 
 ω
 
 i
 
 
 
 
 {\displaystyle \sum _{i=0}^{n-1}X_{i}\omega ^{i}}
 
 where 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 is a <a href="/facts/Primitive_nth_root_of_unity/EqH0ho1Y">primitive nth root of unity</a>. It is the resolvent invariant of a Galois resolvent for the identity group.</li>
<li>A relative resolvent is defined similarly as a resolvent, but considering only the action of the elements of a given subgroup H of Sn, having the property that, if a relative resolvent for a subgroup G of H has a rational simple root and the Galois group of f is contained in H, then the Galois group of f is contained in G. In this context, a usual resolvent is called an absolute resolvent.</li></ul>
<h2 id="resolvent-method">Resolvent method</h2>
The Galois group of a polynomial of degree 
 
 
 
 n
 
 
 {\displaystyle n}
 
 is 
 
 
 
 
 S
 
 n
 
 
 
 
 {\displaystyle S_{n}}
 
 or a <a href="/facts/Proper_subgroup/r91mBgpa">proper subgroup</a> of it. If a polynomial is separable and irreducible, then the corresponding Galois group is a transitive subgroup.
Transitive subgroups of 
 
 
 
 
 S
 
 n
 
 
 
 
 {\displaystyle S_{n}}
 
 form a directed graph: one group can be a subgroup of several groups. One resolvent can tell if the Galois group of a polynomial is a (not necessarily proper) subgroup of given group. The resolvent method is just a systematic way to check groups one by one until only one group is possible. This does not mean that every group must be checked: every resolvent can cancel out many possible groups. For example, for degree five polynomials there is never need for a resolvent of 
 
 
 
 
 D
 
 5
 
 
 
 
 {\displaystyle D_{5}}
 
: resolvents for 
 
 
 
 
 A
 
 5
 
 
 
 
 {\displaystyle A_{5}}
 
 and 
 
 
 
 
 M
 
 20
 
 
 
 
 {\displaystyle M_{20}}
 
 give desired information.
One way is to begin from maximal (transitive) subgroups until the right one is found and then continue with maximal subgroups of that.

<ul><li><a href="/facts/Leonard_Eugene_Dickson/zJ3rPHDJ">Dickson, Leonard E.</a> (1959). Algebraic Theories. New York: Dover Publications Inc. p. ix+276. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-49573-6. {{cite book}}: ISBN / Date incompatibility (help)</li>
<li>Girstmair, K. (1983). "On the computation of resolvents and Galois groups". Manuscripta Mathematica. 43 (2–3): 289–307. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01165834">10.1007/BF01165834</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:123752910">123752910</a>.</li></ul>

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

<ol>
<li id="fn:1">"Resultants, Resolvents and the Computation of Galois Groups" (PDF). Archived from the original (PDF) on 2003-04-04. <a href="https://web.archive.org/web/20030404083026/http://alexhealy.net:80/papers/math250a.pdf" target="_blank">https://web.archive.org/web/20030404083026/http://alexhealy.net:80/papers/math250a.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Resolvent (Galois theory) open-in-new

Resolvent (Galois theory)