For polynomial equations {Abelian equation}, all roots are square roots or first-power roots. Square roots and first-power roots can interchange.
Some polynomial equations are solvable by algebraic operations {Galois theory}, and mathematical groups show which types.
For equations with prime-number degree {Galoisian equation}, solutions are rational functions of two roots, found from two linear equations, two quadratic equations, or one linear and one quadratic equation.
To find polynomial roots, find successively smaller groups down to all smallest normal subgroups {composition series}. The set of composition-series subgroups is unique. If composition-series subgroup indices are prime numbers, roots have radicals.
Quotient groups do not depend on composition-series subgroups {Jordan-Holder theorem}.
To understand critical phenomena, use reiterative group theory {renormalization group theory}. Critical phenomena include Curie point, critical temperature, critical pressure, superconductivity transition temperature, and alloy metal-atom order-disorder temperature. Renormalization-group-theory operations eliminate infinities by reiteration to eliminate higher-order terms.
The non-commutative GL(n) subgroup has matrix operations {representation theory}.
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Date Modified: 2022.0225