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Casus irreducibilis

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Dec 16, 2021

In algebra, casus irreducibilis (Latin for “the irreducible case”) is one of the cases that may arise in solving polynomials of degree 3 or higher with integer 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 roots, which was proven by Pierre Wantzel in 1843.[1] One can see whether a given cubic polynomial is in so-called casus irreducibilis by looking at the discriminant, via Cardano’s formula.[2]

. . . Casus irreducibilis . . .

Let

ax3+bx2+cx+d=0{displaystyle ax^{3}+bx^{2}+cx+d=0}

be a cubic equation with

a0{displaystyle aneq 0}

. Then the discriminant is given by

D:=((x1x2)(x1x3)(x2x3))2=18abcd4ac327a2d2+b2c24b3d .{displaystyle D:={bigl (}(x_{1}-x_{2})(x_{1}-x_{3})(x_{2}-x_{3}){bigr )}^{2}=18abcd-4ac^{3}-27a^{2}d^{2}+b^{2}c^{2}-4b^{3}d~.}

It appears in the algebraic solution and is the square of the product

Δ:=jk(xjxk)=(x1x2)(x1x3)(x2x3)(=±D){displaystyle Delta

 :=prod _{j

of the

(32)=3{displaystyle {tbinom {3}{2}}=3}

differences of the 3 roots

x1,x2,x3{displaystyle x_{1},x_{2},x_{3}}

.[3]

  1. If D , then the polynomial has one real root and two complex non-real roots.
    ΔiR×{displaystyle Delta in imathbb {R} ^{times }}

    is purely imaginary.
    Although there are cubic polynomials with negative discriminant which are irreducible in the modern sense, casus irreducibilis does not apply.[4]

  2. If D = 0, then
    Δ=0{displaystyle Delta =0}

    and there are three real roots; two of them are equal. Whether D = 0 can be found out by the Euclidean algorithm, and if so, the roots by the quadratic formula. Moreover, all roots are real and expressible by real radicals.
    All the cubic polynomials with zero discriminant are reducible.

  3. If D > 0, then
    ΔR×{displaystyle Delta in mathbb {R} ^{times }}

    is non-zero and real, and there are three distinct real roots which are sums of two complex conjugates.
    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.[5]

. . . Casus irreducibilis . . .

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. . . Casus irreducibilis . . .