Editing Group (mathematics) (section)

=== Galois groups ===
{{Main|Galois group}}
''Galois groups'' were developed to help solve polynomial equations by capturing their symmetry features.{{sfn|Robinson|1996|p=viii}}{{sfn|Artin|1998}} For example, the solutions of the [[quadratic equation]] <math>ax^2+bx+c=0</math> are given by
<math display=block alt="x = (negative b plus or minus the squareroot of (b squared minus 4 a c)) over 2a">x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}.</math>
Each solution can be obtained by replacing the <math>\pm</math> sign by <math>+</math> or {{tmath|1= - }}; analogous formulae are known for [[cubic equation|cubic]] and [[quartic equation]]s, but do ''not'' exist in general for [[quintic equation|degree 5]] and higher.{{sfn|Lang|2002|loc=Chapter VI (see in particular p. 273 for concrete examples)}} In the [[quadratic formula]], changing the sign (permuting the resulting two solutions) can be viewed as a (very simple) group operation. Analogous Galois groups act on the solutions of higher-degree polynomial equations and are closely related to the existence of formulas for their solution. Abstract properties of these groups (in particular their [[solvable group|solvability]]) give a criterion for the ability to express the solutions of these polynomials using solely addition, multiplication, and [[Nth root|roots]] similar to the formula above.{{sfn|Lang|2002|p=292|loc=(Theorem VI.7.2)}}

Modern [[Galois theory]] generalizes the above type of Galois groups by shifting to field theory and considering [[field extension]]s formed as the [[splitting field]] of a polynomial. This theory establishes—via the [[fundamental theorem of Galois theory]]—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.{{sfn|Stewart|2015|loc=§12.1}}