References analysis of the Wikipedia article "Casus irreducibilis" in the English version

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James Pierpont in Annals of Mathematics 1900-1901 on p. 38: "To Cardan and his contemporaries who had no idea how such cube roots could be found this case was highly paradoxical. Since that time mathematicians have attempted to present these real roots as sums of real radicals. As their efforts were unsuccessful, the case when D > 0 was known as the casus irreducibilis."
Artur Ekert Complex and unpredictable Cardano takes Cardano’s example

x^{3}-15x-4=0

having

{\textstyle q^{2}/4+p^{3}/27=(-4)^{2}/4+(-15)^{3}/27=-121<0}

and writes on p. 9: "Cardano knew that

x=4

was one of the solutions and yet it was a casus irreducibilis". This shows that in the 16th century "irreducibilis" must have meant something like "not reducible to real radicals".
On the other hand, Cardano’s example may be used to show how real roots can arise from cube roots of non-real numbers:

We have	$p=-15,\;q=-4$ ,
which yields	$d:={q^{2} \over 4}+{p^{3} \over 27}=-121$ ,
from which	$-{q \over 2}\pm {\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}=-{q \over 2}\pm {\sqrt {d}}=2\pm 11\,i$ .
In the 16th century it was difficult („verè sophistica“) to find that
	${\sqrt[{3}]{2+11^{\color {white};}i}}=2+i=:u$
and	${\sqrt[{3}]{2-11^{\color {white};}i}}=2-i=:v$ ,
so that	$t_{k}=\omega _{k}{\sqrt[{3}]{-{q \over 2}+{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}+\omega _{k}^{2}{\sqrt[{3}]{-{q \over 2}-{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}$
	$=\omega _{k}u+\omega _{k}^{2}v$ .
This means in detail:
1st root	$t_{1}=\omega _{1}u+\omega _{1}^{2}v$
	$=1\cdot (2+i)+1\cdot (2-i)$
	$=4$ ,
2nd root	$t_{2}=\omega _{2}u+\omega _{2}^{2}v$
	$={\biggl (}\!\!-{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i{\biggr )}\cdot (2+i)+{\biggl (}\!\!-{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i{\biggr )}\cdot (2-i)$
	$=-2-{\sqrt {3}}$ ,
3rd root	$t_{3}=\omega _{3}u+\omega _{3}^{2}v$
	$={\biggl (}\!\!-{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i{\biggr )}\cdot (2+i)+{\biggl (}\!\!-{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i{\biggr )}\cdot (2-i)$
	$=-2+{\sqrt {3}}$ .

It may be noticed that ${\textstyle d:=q^{2}/4+p^{3}/27=-121}$ is not the discriminant $D$ ; it is $D=-108\,d=13068=2^{2}3^{3}11^{2}$ with the sign inverted. Interestingly ${\textstyle {\sqrt {d}}=i{\sqrt {D/108}}}$ occurs in Cardano’s formula (as well as the primitive 3rd roots of unity $\omega _{2,3}$ with their $i{\sqrt {3}}$ ), although ${\sqrt {D}}~,$ and not ${\sqrt {d}}~,$ is necessarily an element of the splitting field.

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Cox (2012), Section 1.3B Trigonometric Solution of the Cubic, pp. 18–19. Cox, David A. (2012), Galois Theory, Pure and Applied Mathematics (2nd ed.), John Wiley & Sons, doi:10.1002/9781118218457, ISBN 978-1-118-07205-9. See in particular Section 1.3 Cubic Equations over the Real Numbers (pp. 15–22) and Section 8.6 The Casus Irreducibilis (pp. 220–227).

Cox (2012), Theorem 8.6.5, p. 222. Cox, David A. (2012), Galois Theory, Pure and Applied Mathematics (2nd ed.), John Wiley & Sons, doi:10.1002/9781118218457, ISBN 978-1-118-07205-9. See in particular Section 1.3 Cubic Equations over the Real Numbers (pp. 15–22) and Section 8.6 The Casus Irreducibilis (pp. 220–227).

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Casus irreducibilis (English Wikipedia)

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