In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, ''casus irreducibilis'' (
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
for "the irreducible case") is one of the cases that may arise in solving polynomials of
degree 3 or higher with
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
coefficients algebraically (as opposed to numerically), 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
Real may refer to:
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* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
roots, which was proven by
Pierre Wantzel
Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge.
In a paper from 1837, Wantzel pr ...
in 1843.
One can see whether a given cubic polynomial is in so-called ''casus irreducibilis'' by looking at the
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
, via
Cardano's formula
In algebra, a cubic equation in one variable is an equation of the form
:ax^3+bx^2+cx+d=0
in which is nonzero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the ...
.
The three cases of the discriminant
Let
:
be a cubic equation with
. Then the discriminant is given by
:
It appears in the algebraic solution and is the square of the product
:
of the
differences of the 3 roots
.
# If , then the polynomial has one real root and two complex non-real roots.
is purely imaginary.
Although there are cubic polynomials with negative discriminant which are irreducible in the modern sense, ''casus irreducibilis'' does not apply.
# If , then
and there are three real roots; two of them are equal. Whether can be found out by the
Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...
, and if so, the roots by the
quadratic formula
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 (direct factoring, gr ...
. Moreover, all roots are real and expressible by real radicals.
All the cubic polynomials with zero discriminant are reducible.
# If , then
is non-zero and real, and there are three distinct real roots which are sums of two
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
s.
Because they require complex numbers (in the understanding of the time: cube roots from non-real numbers, i.e. from square roots from negative numbers) to express them in radicals, this case in the 16th century has been termed ''casus irreducibilis''.
Formal statement and proof
More generally, suppose that is a
formally real field
In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field.
Alternative definitions
The definition given above i ...
, and that is a cubic polynomial, irreducible over , but having three real roots (roots in the
real closure of ). Then ''casus irreducibilis'' states that it is impossible to express a solution of by radicals with radicands .
To prove this, note that the discriminant is positive. Form the
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
. Since this is or a
quadratic extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
of (depending in whether or not is a square in ), remains irreducible in it. Consequently, the
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
of over is the cyclic group . Suppose that can be solved by real radicals. Then can be
split
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Arts, enterta ...
by a tower of
cyclic extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable ...
s
:
At the final step of the tower, is irreducible in the penultimate field , but splits in for some . But this is a cyclic field extension, and so must contain a
conjugate of and therefore a
primitive 3rd root of unity.
However, there are no primitive 3rd roots of unity in a real closed field. Suppose that ω is a primitive 3rd root of unity. Then, by the axioms defining an
ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fiel ...
, ω and ω
2 are both positive, because otherwise their cube (=1) would be negative. But if ω
2>ω, then cubing both sides gives 1>1, a contradiction; similarly if ω>ω
2.
Solution in non-real radicals
Cardano's solution
The equation can be depressed to a
monic trinomial
In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials.
Examples of trinomial expressions
# 3x + 5y + 8z with x, y, z variables
# 3t + 9s^2 + 3y^3 with t, s, y variables
# 3ts + 9t + 5s with t, s variables
# ...
by dividing by
and substituting (the
Tschirnhaus transformation
In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683.
Simply, it is a method for transforming a polynomial equation ...
), giving the equation where
:
:
Then regardless of the number of real roots, by
Cardano's solution the three roots are given by
: