Solution in radicals

Last updated May 03, 2023

A solution in radicals or algebraic solution is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of a polynomial equation, and relies only on addition, subtraction, multiplication, division, raising to integer powers, and the extraction of $n$ th roots (square roots, cube roots, and other integer roots).

A well-known example is the solution

x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}

of the quadratic equation

ax^{2}+bx+c=0.

There exist more complicated algebraic solutions for cubic equations ^[1] and quartic equations.^[2] The Abel–Ruffini theorem,^[3]^: 211 and, more generally Galois theory, state that some quintic equations, such as

x^{5}-x+1=0,

do not have any algebraic solution. The same is true for every higher degree. However, for any degree there are some polynomial equations that have algebraic solutions; for example, the equation $x^{10}=2$ can be solved as $x=\pm {\sqrt[{10}]{2}}.$ The eight other solutions are nonreal complex numbers, which are also algebraic and have the form $x=\pm r{\sqrt[{10}]{2}},$ where $r$ is a fifth root of unity, which can be expressed with two nested square roots. See also Quintic function § Other solvable quintics for various other examples in degree 5.

Évariste Galois introduced a criterion allowing one to decide which equations are solvable in radicals. See Radical extension for the precise formulation of his result.

Algebraic solutions form a subset of closed-form expressions, because the latter permit transcendental functions (non-algebraic functions) such as the exponential function, the logarithmic function, and the trigonometric functions and their inverses.

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In mathematics, a polynomial is an expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate $x$ is $x 2 - 4 x + 7$ . An example with three indeterminates is $x 3 + 2 xyz 2 - yz + 1$ .

In algebra, a quadratic equation is any equation that can be rearranged in standard form as

In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring, completing the square, graphing and others.

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In mathematics, the Abel–Ruffini theorem states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, general means that the coefficients of the equation are viewed and manipulated as indeterminates.

<span class="mw-page-title-main">Root of unity</span> Number that has an integer power equal to 1

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power $n$ . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.

<span class="mw-page-title-main">Cubic equation</span> Polynomial equation of degree 3

In algebra, a cubic equation in one variable is an equation of the form

<span class="mw-page-title-main">Quartic equation</span> Polynomial equation

In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is

In mathematics, a quintic function is a function of the form

<span class="mw-page-title-main">Quartic function</span> Polynomial function of degree four

In algebra, a quartic function is a function of the form

In mathematics, an algebraic equation or polynomial equation is an equation of the form

In algebra, the theory of equations is the study of algebraic equations, which are equations defined by a polynomial. The main problem of the theory of equations was to know when an algebraic equation has an algebraic solution. This problem was completely solved in 1830 by Évariste Galois, by introducing what is now called Galois theory.

In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations, and functions, but usually no limit, or integral.

In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are:

In algebra, the Bring radical or ultraradical of a real number a is the unique real root of the polynomial

In mathematics, an algebraic expression is an expression built up from constant algebraic numbers, variables, and the algebraic operations. For example, $3 x 2 - 2 xy + c$ is an algebraic expression. Since taking the square root is the same as raising to the power 1/2, the following is also an algebraic expression:

In algebra, a sexticpolynomial is a polynomial of degree six. A sextic equation is a polynomial equation of degree six—that is, an equation whose left hand side is a sextic polynomial and whose right hand side is zero. More precisely, it has the form:

In algebra, casus irreducibilis is one of the cases that may arise in solving polynomials of degree 3 or higher with integer coefficients algebraically, i.e., by obtaining roots that are expressed with radicals. It shows that many algebraic numbers are real-valued but cannot be expressed in radicals without introducing complex numbers. The most notable occurrence of casus irreducibilis is in the case of cubic polynomials that have three real roots, which was proven by Pierre Wantzel in 1843. One can see whether a given cubic polynomial is in so-called casus irreducibilis by looking at the discriminant, via Cardano's formula.

In algebra, a septic equation is an equation of the form

References

↑ Nickalls, R. W. D., "A new approach to solving the cubic: Cardano's solution revealed," Mathematical Gazette 77, November 1993, 354-359.
↑ Carpenter, William, "On the solution of the real quartic," Mathematics Magazine 39, 1966, 28-30.
↑ Jacobson, Nathan (2009), Basic Algebra 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1

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[1] Nickalls, R. W. D., "A new approach to solving the cubic: Cardano's solution revealed," Mathematical Gazette 77, November 1993, 354-359.

[2] Carpenter, William, "On the solution of the real quartic," Mathematics Magazine 39, 1966, 28-30.

[3] Jacobson, Nathan (2009), Basic Algebra 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1

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