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More generally, in the context of [[polynomial equation]]s, a closed form of a solution is a [[solution in radicals]]; that is, a closed-form expression for which the allowed functions are only {{mvar|n}}th-roots and field operations (+, -, /, *). In fact, [[field theory (mathematics)|field theory]] allows showing that if a solution of a polynomial equation has a closed form involving exponentials, logarithms or trigonometric functions, then it has also a closed form that does not involve these functions.{{cn|date=August 2023}}
There are expressions in radicals for all solutions of [[cubic equation]]s (degree 3) and [[quartic equation]]s (degree 4).
In higher degrees, [[Abel–Ruffini theorem]] states that there are equations whose solutions cannot be expressed in radicals, and, thus, have no closed forms. The simplest example is the equation <math>x^5-x+1.</math> [[Galois theory]] provides an [[algorithmic method]] for deciding whether a particular polynomial equation can be solved in radicals.
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