Solvability by radicals

Solvability by radicals is the possibility of expressing the roots of a polynomial using field operations and finitely many extractions of roots.

Galois theory: The study of relationships between field extensions and groups of symmetries of algebraic equations. It connects radical expressions for roots to the structure of their symmetry groups.
Polynomial: An expression in variables formed from coefficients using addition, subtraction, multiplication, and nonnegative integer powers. Solvability by radicals asks whether a polynomial’s roots have a particular kind of expression.
Quadratic formula: A formula expressing the roots of a quadratic polynomial using its coefficients and a square root. It gives a universal radical formula for polynomials of degree two.
Gerolamo Cardano: An Italian mathematician whose 1545 book Ars Magna published methods for solving cubic equations. His account helped establish explicit radical formulas for cubic equations.
Numerical root-finding: Methods for approximating solutions of equations with numerical values. A polynomial can lack radical solutions while its roots remain numerically approximable.
Solvable group: A group whose derived series eventually reaches the trivial group. A polynomial is solvable by radicals exactly when its Galois group is solvable, in characteristic zero.
Root of a polynomial: A value that makes a polynomial evaluate to zero. The objects to be expressed by radicals are the polynomial’s roots.
Cubic formula: A formula expressing the roots of a general cubic using its coefficients and radicals. Cubic equations remain solvable by radicals, though their formulas can involve complex intermediate values.
Lodovico Ferrari: An Italian mathematician who discovered a method for solving quartic equations by radicals. His method extended explicit radical solution from cubics to quartics.
Implicit function: A function specified by an equation it satisfies rather than by an explicit formula. A root can be defined exactly by its polynomial even when radicals cannot express it.

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