Resolvent (Galois theory)

In <a href="/facts/Galois_theory/54wscJfj">Galois theory</a>, a discipline within the field of <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, a resolvent for a <a href="/facts/Permutation_group/vRsNNwwx">permutation group</a> G is a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> whose <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a> depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a <a href="/facts/Rational_number/VJQmyY05">rational</a> <a href="/facts/Root_of_a_polynomial/mlK3dhoT">root</a> <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the <a href="/facts/Galois_group/A0p35p5c">Galois group</a> of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the <a href="/facts/Converse_(logic)/VhlUrxxx">converse</a> is true if the rational root is a <a href="/facts/Simple_root_(polynomial)/Lzak8VVx">simple root</a>.
Resolvents were introduced by <a href="/facts/Joseph_Louis_Lagrange/bTQUaSTv">Joseph Louis Lagrange</a> and systematically used by <a href="/facts/%25C3%2589variste_Galois/T0BPM1zP">Évariste Galois</a>. Nowadays they are still a fundamental tool to compute <a href="/facts/Galois_group/A0p35p5c">Galois groups</a>. The simplest examples of resolvents are

<ul><li>
 
 
 
 
 X
 
 2
 
 
 −
 Δ
 
 
 {\displaystyle X^{2}-\Delta }
 
 where 
 
 
 
 Δ
 
 
 {\displaystyle \Delta }
 
 is the <a href="/facts/Discriminant/0SmQtrvU">discriminant</a>, which is a resolvent for the <a href="/facts/Alternating_group/yethKwyo">alternating group</a>. In the case of a <a href="/facts/Cubic_equation/Eqm3HCMx">cubic equation</a>, this resolvent is sometimes called the quadratic resolvent; its roots appear explicitly in the formulas for the roots of a cubic equation.</li>
<li>The <a href="/facts/Resolvent_cubic/cxbIHK9s">cubic resolvent</a> of a <a href="/facts/Quartic_function/lHuBhMk7">quartic equation</a>, which is a resolvent for the <a href="/facts/Dihedral_group/1PUlRh4M">dihedral group</a> of 8 elements.</li>
<li>The <a href="/facts/Quintic_function/oeNDGj9l">Cayley resolvent</a> is a resolvent for the maximal <a href="/facts/Solvable_group/Na7lKIns">solvable</a> Galois group in degree five. It is a polynomial of <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> 6.</li></ul>
These three resolvents have the property of being always separable, which means that, if they have a <a href="/facts/Multiple_root/dndUyetA">multiple root</a>, then the polynomial p is not <a href="/facts/Irreducible_polynomial/PTeuKm4W">irreducible</a>. It is not known if there is an always separable resolvent for every group of permutations.
For every equation the roots may be expressed in terms of <a href="/facts/Nth_root/XrRjtJGe">radicals</a> and of a root of a resolvent for a solvable group, because the Galois group of the equation over the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> generated by this root is solvable.

Resolvent (Galois theory) open-in-new

Resolvent (Galois theory)