Resolvent cubic

Last updated February 22, 2024

In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four:

The coefficients of the resolvent cubic can be obtained from the coefficients of $P (x)$ using only sums, subtractions and multiplications.
Knowing the roots of the resolvent cubic of $P (x)$ is useful for finding the roots of $P (x)$ itself. Hence the name “resolvent cubic”.
The polynomial $P (x)$ has a multiple root if and only if its resolvent cubic has a multiple root.

Definitions

Suppose that the coefficients of $P (x)$ belong to a field $k$ whose characteristic is different from $2$ . In other words, we are working in a field in which $1 + 1 \neq 0$ . Whenever roots of $P (x)$ are mentioned, they belong to some extension $K$ of $k$ such that $P (x)$ factors into linear factors in $K [x]$ . If $k$ is the field $Q$ of rational numbers, then $K$ can be the field $C$ of complex numbers or the field $Q$ of algebraic numbers.

In some cases, the concept of resolvent cubic is defined only when $P (x)$ is a quartic in depressed form—that is, when $a 3 = 0$ .

Note that the fourth and fifth definitions below also make sense and that the relationship between these resolvent cubics and $P (x)$ are still valid if the characteristic of $k$ is equal to $2$ .

First definition

Suppose that $P (x)$ is a depressed quartic—that is, that $a 3 = 0$ . A possible definition of the resolvent cubic of $P (x)$ is:^[1]

R_{1}(y)=8y^{3}+8a_{2}y^{2}+(2{a_{2}}^{2}-8a_{0})y-{a_{1}}^{2}.

The origin of this definition lies in applying Ferrari's method to find the roots of $P (x)$ . To be more precise:

{\begin{aligned}P(x)=0&\Longleftrightarrow x^{4}+a_{2}x^{2}=-a_{1}x-a_{0}\\&\Longleftrightarrow \left(x^{2}+{\frac {a_{2}}{2}}\right)^{2}=-a_{1}x-a_{0}+{\frac {{a_{2}}^{2}}{4}}.\end{aligned}}

Add a new unknown, $y$ , to $x 2 + a 2 /2$ . Now you have:

{\begin{aligned}\left(x^{2}+{\frac {a_{2}}{2}}+y\right)^{2}&=-a_{1}x-a_{0}+{\frac {{a_{2}}^{2}}{4}}+2x^{2}y+a_{2}y+y^{2}\\&=2yx^{2}-a_{1}x-a_{0}+{\frac {{a_{2}}^{2}}{4}}+a_{2}y+y^{2}.\end{aligned}}

If this expression is a square, it can only be the square of

{\sqrt {2y}}\,x-{\frac {a_{1}}{2{\sqrt {2y}}}}.

But the equality

\left({\sqrt {2y}}\,x-{\frac {a_{1}}{2{\sqrt {2y}}}}\right)^{2}=2yx^{2}-a_{1}x-a_{0}+{\frac {{a_{2}}^{2}}{4}}+a_{2}y+y^{2}

is equivalent to

{\frac {{a_{1}}^{2}}{8y}}=-a_{0}+{\frac {{a_{2}}^{2}}{4}}+a_{2}y+y^{2}{\text{,}}

and this is the same thing as the assertion that $R 1 (y)$ = 0.

If $y 0$ is a root of $R 1 (y)$ , then it is a consequence of the computations made above that the roots of $P (x)$ are the roots of the polynomial

x^{2}-{\sqrt {2y_{0}}}\,x+{\frac {a_{2}}{2}}+y_{0}+{\frac {a_{1}}{2{\sqrt {2y_{0}}}}}

together with the roots of the polynomial

x^{2}+{\sqrt {2y_{0}}}\,x+{\frac {a_{2}}{2}}+y_{0}-{\frac {a_{1}}{2{\sqrt {2y_{0}}}}}.

Of course, this makes no sense if $y 0 = 0$ , but since the constant term of $R 1 (y)$ is $- a 12$ , $0$ is a root of $R 1 (y)$ if and only if $a 1 = 0$ , and in this case the roots of $P (x)$ can be found using the quadratic formula.

Second definition

Another possible definition^[1] (still supposing that $P (x)$ is a depressed quartic) is

R_{2}(y)=8y^{3}-4a_{2}y^{2}-8a_{0}y+4a_{2}a_{0}-{a_{1}}^{2}

The origin of this definition is similar to the previous one. This time, we start by doing:

{\begin{aligned}P(x)=0&\Longleftrightarrow x^{4}=-a_{2}x^{2}-a_{1}x-a_{0}\\&\Longleftrightarrow (x^{2}+y)^{2}=-a_{2}x^{2}-a_{1}x-a_{0}+2yx^{2}+y^{2}\end{aligned}}

and a computation similar to the previous one shows that this last expression is a square if and only if

8y^{3}-4a_{2}y^{2}-8a_{0}y+4a_{2}a_{0}-{a_{1}}^{2}=0{\text{.}}

A simple computation shows that

R_{2}\left(y+{\frac {a_{2}}{2}}\right)=R_{1}(y).

Third definition

Another possible definition^[2]^[3] (again, supposing that $P (x)$ is a depressed quartic) is

R_{3}(y)=y^{3}+2a_{2}y^{2}+({a_{2}}^{2}-4a_{0})y-{a_{1}}^{2}{\text{.}}

The origin of this definition lies in another method of solving quartic equations, namely Descartes' method. If you try to find the roots of $P (x)$ by expressing it as a product of two monic quadratic polynomials $x 2 + αx + β$ and $x 2 - αx + γ$ , then

P(x)=(x^{2}+\alpha x+\beta )(x^{2}-\alpha x+\gamma )\Longleftrightarrow \left\{{\begin{array}{l}\beta +\gamma -\alpha ^{2}=a_{2}\\\alpha (-\beta +\gamma )=a_{1}\\\beta \gamma =a_{0}.\end{array}}\right.

If there is a solution of this system with $α \neq 0$ (note that if $a 1 \neq 0$ , then this is automatically true for any solution), the previous system is equivalent to

\left\{{\begin{array}{l}\beta +\gamma =a_{2}+\alpha ^{2}\\-\beta +\gamma ={\frac {a_{1}}{\alpha }}\\\beta \gamma =a_{0}.\end{array}}\right.

It is a consequence of the first two equations that then

\beta ={\frac {1}{2}}\left(a_{2}+\alpha ^{2}-{\frac {a_{1}}{\alpha }}\right)

and

\gamma ={\frac {1}{2}}\left(a_{2}+\alpha ^{2}+{\frac {a_{1}}{\alpha }}\right).

After replacing, in the third equation, $β$ and $γ$ by these values one gets that

\left(a_{2}+\alpha ^{2}\right)^{2}-{\frac {{a_{1}}^{2}}{\alpha ^{2}}}=4a_{0}{\text{,}}

and this is equivalent to the assertion that $α 2$ is a root of $R 3 (y)$ . So, again, knowing the roots of $R 3 (y)$ helps to determine the roots of $P (x)$ .

Note that

R_{3}(y)=R_{1}\left({\frac {y}{2}}\right){\text{.}}

Fourth definition

Still another possible definition is^[4]

R_{4}(y)=y^{3}-a_{2}y^{2}+(a_{1}a_{3}-4a_{0})y+4a_{0}a_{2}-{a_{1}}^{2}-a_{0}{a_{3}}^{2}.

In fact, if the roots of $P (x)$ are $α 1, α 2, α 3$ , and $α 4$ , then

R_{4}(y)={\bigl (}y-(\alpha _{1}\alpha _{2}+\alpha _{3}\alpha _{4}){\bigr )}{\bigl (}y-(\alpha _{1}\alpha _{3}+\alpha _{2}\alpha _{4}){\bigr )}{\bigl (}y-(\alpha _{1}\alpha _{4}+\alpha _{2}\alpha _{3}){\bigr )}{\text{,}}

a fact the follows from Vieta's formulas. In other words, R₄(y) is the monic polynomial whose roots are $α 1 α 2 + α 3 α 4$ , $α 1 α 3 + α 2 α 4$ , and $α 1 α 4 + α 2 α 3$ .

It is easy to see that

\alpha _{1}\alpha _{2}+\alpha _{3}\alpha _{4}-(\alpha _{1}\alpha _{3}+\alpha _{2}\alpha _{4})=(\alpha _{1}-\alpha _{4})(\alpha _{2}-\alpha _{3}){\text{,}}

\alpha _{1}\alpha _{3}+\alpha _{2}\alpha _{4}-(\alpha _{1}\alpha _{4}+\alpha _{2}\alpha _{3})=(\alpha _{1}-\alpha _{2})(\alpha _{3}-\alpha _{4}){\text{,}}

\alpha _{1}\alpha _{2}+\alpha _{3}\alpha _{4}-(\alpha _{1}\alpha _{4}+\alpha _{2}\alpha _{3})=(\alpha _{1}-\alpha _{3})(\alpha _{2}-\alpha _{4}){\text{.}}

Therefore, $P (x)$ has a multiple root if and only if $R 4 (y)$ has a multiple root. More precisely, $P (x)$ and $R 4 (y)$ have the same discriminant.

One should note that if $P (x)$ is a depressed polynomial, then

{\begin{aligned}R_{4}(y)&=y^{3}-a_{2}y^{2}-4a_{0}y+4a_{0}a_{2}-{a_{1}}^{2}\\&=R_{2}\left({\frac {y}{2}}\right){\text{.}}\end{aligned}}

Fifth definition

Yet another definition is^[5]^[6]

R_{5}(y)=y^{3}-2a_{2}y^{2}+({a_{2}}^{2}+a_{3}a_{1}-4a_{0})y+{a_{1}}^{2}-a_{3}a_{2}a_{1}+{a_{3}}^{2}a_{0}{\text{.}}

If, as above, the roots of $P (x)$ are $α 1, α 2, α 3$ , and $α 4$ , then

R_{5}(y)={\bigl (}y-(\alpha _{1}+\alpha _{2})(\alpha _{3}+\alpha _{4}){\bigr )}{\bigl (}y-(\alpha _{1}+\alpha _{3})(\alpha _{2}+\alpha _{4}){\bigr )}{\bigl (}y-(\alpha _{1}+\alpha _{4})(\alpha _{2}+\alpha _{3}){\bigr )}{\text{,}}

again as a consequence of Vieta's formulas. In other words, $R 5 (y)$ is the monic polynomial whose roots are $(α 1 + α 2)(α 3 + α 4)$ , $(α 1 + α 3)(α 2 + α 4)$ , and $(α 1 + α 4)(α 2 + α 3)$ .

It is easy to see that

(\alpha _{1}+\alpha _{2})(\alpha _{3}+\alpha _{4})-(\alpha _{1}+\alpha _{3})(\alpha _{2}+\alpha _{4})=-(\alpha _{1}-\alpha _{4})(\alpha _{2}-\alpha _{3}){\text{,}}

(\alpha _{1}+\alpha _{2})(\alpha _{3}+\alpha _{4})-(\alpha _{1}+\alpha _{4})(\alpha _{2}+\alpha _{3})=-(\alpha _{1}-\alpha _{3})(\alpha _{2}-\alpha _{4}){\text{,}}

(\alpha _{1}+\alpha _{3})(\alpha _{2}+\alpha _{4})-(\alpha _{1}+\alpha _{4})(\alpha _{2}+\alpha _{3})=-(\alpha _{1}-\alpha _{2})(\alpha _{3}-\alpha _{4}){\text{.}}

Therefore, as it happens with $R 4 (y)$ , $P (x)$ has a multiple root if and only if $R 5 (y)$ has a multiple root. More precisely, $P (x)$ and $R 5 (y)$ have the same discriminant. This is also a consequence of the fact that $R 5 (y + a 2)$ = $- R 4 (- y)$ .

Note that if $P (x)$ is a depressed polynomial, then

{\begin{aligned}R_{5}(y)&=y^{3}-2a_{2}y^{2}+({a_{2}}^{2}-4a_{0})y+{a_{1}}^{2}\\&=-R_{3}(-y)\\&=-R_{1}\left(-{\frac {y}{2}}\right){\text{.}}\end{aligned}}

Applications

Solving quartic equations

It was explained above how $R 1 (y)$ , $R 2 (y)$ , and $R 3 (y)$ can be used to find the roots of $P (x)$ if this polynomial is depressed. In the general case, one simply has to find the roots of the depressed polynomial $P (x - a 3 /4)$ . For each root $x 0$ of this polynomial, $x 0 - a 3 /4$ is a root of $P (x)$ .

Factoring quartic polynomials

If a quartic polynomial $P (x)$ is reducible in $k [x]$ , then it is the product of two quadratic polynomials or the product of a linear polynomial by a cubic polynomial. This second possibility occurs if and only if $P (x)$ has a root in $k$ . In order to determine whether or not $P (x)$ can be expressed as the product of two quadratic polynomials, let us assume, for simplicity, that $P (x)$ is a depressed polynomial. Then it was seen above that if the resolvent cubic $R 3 (y)$ has a non-null root of the form $α 2$ , for some $α \in k$ , then such a decomposition exists.

This can be used to prove that, in $R [x]$ , every quartic polynomial without real roots can be expressed as the product of two quadratic polynomials. Let $P (x)$ be such a polynomial. We can assume without loss of generality that $P (x)$ is monic. We can also assume without loss of generality that it is a reduced polynomial, because $P (x)$ can be expressed as the product of two quadratic polynomials if and only if $P (x - a 3 /4)$ can and this polynomial is a reduced one. Then $R 3 (y)$ = $y 3 + 2 a 2 y 2 + (a 22 - 4 a 0) y - a 12$ . There are two cases:

If $a 1 \neq 0$ then $R 3 (0)$ = $- a 12 < 0$ . Since $R 3 (y) > 0$ if $y$ is large enough, then, by the intermediate value theorem, $R 3 (y)$ has a root $y 0$ with $y 0 > 0$ . So, we can take $α$ = $\sqrt y 0$ .
If $a 1$ = $0$ , then $R 3 (y)$ = $y 3 + 2 a 2 y 2 + (a 22 - 4 a 0) y$ . The roots of this polynomial are $0$ and the roots of the quadratic polynomial $y 2 + 2 a 2 y + a 22 - 4 a 0$ . If $a 22 - 4 a 0 < 0$ , then the product of the two roots of this polynomial is smaller than $0$ and therefore it has a root greater than $0$ (which happens to be $- a 2 + 2 \sqrt a 0$ ) and we can take $α$ as the square root of that root. Otherwise, $a 22 - 4 a 0 \geq 0$ and then,

P(x)=\left(x^{2}+{\frac {a_{2}+{\sqrt {{a_{2}}^{2}-4a_{0}}}}{2}}\right)\left(x^{2}+{\frac {a_{2}-{\sqrt {{a_{2}}^{2}-4a_{0}}}}{2}}\right){\text{.}}

More generally, if $k$ is a real closed field, then every quartic polynomial without roots in $k$ can be expressed as the product of two quadratic polynomials in $k [x]$ . Indeed, this statement can be expressed in first-order logic and any such statement that holds for $R$ also holds for any real closed field.

A similar approach can be used to get an algorithm^[2] to determine whether or not a quartic polynomial $P (x) \in Q [x]$ is reducible and, if it is, how to express it as a product of polynomials of smaller degree. Again, we will suppose that $P (x)$ is monic and depressed. Then $P (x)$ is reducible if and only if at least one of the following conditions holds:

The polynomial $P (x)$ has a rational root (this can be determined using the rational root theorem).
The resolvent cubic $R 3 (y)$ has a root of the form $α 2$ , for some non-null rational number $α$ (again, this can be determined using the rational root theorem).
The number $a 22 - 4 a 0$ is the square of a rational number and $a 1$ = $0$ .

Indeed:

If $P (x)$ has a rational root $r$ , then $P (x)$ is the product of $x - r$ by a cubic polynomial in $Q [x]$ , which can be determined by polynomial long division or by Ruffini's rule.
If there is a rational number $α \neq 0$ such that $α 2$ is a root of $R 3 (y)$ , it was shown above how to express $P (x)$ as the product of two quadratic polynomials in $Q [x]$ .
Finally, if the third condition holds and if $δ \in Q$ is such that $δ 2$ = $a 22 - 4 a 0$ , then $P (x)$ = $(x 2 + (a 2 + δ)/2)(x 2 + (a 2 - δ)/2)$ .

Galois groups of irreducible quartic polynomials

The resolvent cubic of an irreducible quartic polynomial $P (x)$ can be used to determine its Galois group $G$ ; that is, the Galois group of the splitting field of $P (x)$ . Let $m$ be the degree over $k$ of the splitting field of the resolvent cubic (it can be either $R 4 (y)$ or $R 5 (y)$ ; they have the same splitting field). Then the group $G$ is a subgroup of the symmetric group $S 4$ . More precisely:^[4]

If $m = 1$ (that is, if the resolvent cubic factors into linear factors in $k$ ), then $G$ is the group ${e, (12)(34), (13)(24), (14)(23)$ }.
If $m = 2$ (that is, if the resolvent cubic has one and, up to multiplicity, only one root in $k$ ), then, in order to determine $G$ , one can determine whether or not $P (x)$ is still irreducible after adjoining to the field $k$ the roots of the resolvent cubic. If not, then $G$ is a cyclic group of order 4; more precisely, it is one of the three cyclic subgroups of $S 4$ generated by any of its six $4$ -cycles. If it is still irreducible, then $G$ is one of the three subgroups of $S 4$ of order $8$ , each of which is isomorphic to the dihedral group of order $8$ .
If $m = 3$ , then $G$ is the alternating group $A 4$ .
If $m = 6$ , then $G$ is the whole group $S 4$ .

Related Research Articles

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

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 original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry.

In elementary algebra, the quadratic formula is a formula that provides the solutions to a quadratic equation. Other ways of solving a quadratic equation, such as completing the square, yield the same solutions.

<span class="mw-page-title-main">Factorization</span> (Mathematical) decomposition into a product

In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, $3 \times 5$ is an integer factorization of $15$ , and $(x - 2)(x + 2)$ is a polynomial factorization of $x 2 - 4$ .

In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial splits, i.e., decomposes into linear factors.

<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">Quadratic function</span> Polynomial function of degree two

In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before the 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" and "quadratic function" were almost synonymous. This is still the case in many elementary courses, where both terms are often abbreviated as "quadratic".

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

In mathematics, a cubic function is a function of the form $that is, a polynomial function of degree three. In many texts, the coefficients a, b, c, and d are supposed to be real numbers, and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to complex numbers. In other cases, the coefficients may be complex numbers, and the function is a complex function that has the set of the complex numbers as its codomain, even when the domain is restricted to the real numbers.$

In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials $with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irreducible polynomial splits into linear terms when reduced mod . That is, it determines for which prime numbers the relation$

<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 $, where P is a polynomial with coefficients in some field, often the field of the rational numbers. For example, is an algebraic equation with integer coefficients and$

This article describes periodic points of some complex quadratic maps. A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic map is one that involves the previous value raised to the powers one and two; and a complex map is one in which the variable and the parameters are complex numbers. A periodic point of a map is a value of the variable that occurs repeatedly after intervals of a fixed length.

In algebra, a nested radical is a radical expression that contains (nests) another radical expression. Examples include

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

In mathematics, a polynomial transformation consists of computing the polynomial whose roots are a given function of the roots of a polynomial. Polynomial transformations such as Tschirnhaus transformations are often used to simplify the solution of algebraic equations.

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 Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the converse is true if the rational root is a simple root. Resolvents were introduced by Joseph Louis Lagrange and systematically used by Évariste Galois. Nowadays they are still a fundamental tool to compute Galois groups. The simplest examples of resolvents are

Kunerth's algorithm is an algorithm for computing the modular square root of a given number. The algorithm does not require the factorization of the modulus, and relies on modular operations that is often easy when the given number is prime.

References

1 2 Tignol, Jean-Pierre (2016), "Quartic equations", Galois' Theory of algebraic equations (2nd ed.), World Scientific, ISBN 978-981-4704-69-4, Zbl 1333.12001
1 2 Brookfield, G. (2007), "Factoring quartic polynomials: A lost art" (PDF), Mathematics Magazine , 80 (1): 67–70, doi:10.1080/0025570X.2007.11953453, JSTOR 27642994, S2CID 53375377, Zbl 1227.97040, archived from the original (PDF) on 2015-02-21
↑ Hartshorne, Robin (1997), "Construction problems and field extensions: Cubic and quartic equations", Geometry: Euclid and Beyond, Springer-Verlag, ISBN 0-387-98650-2, Zbl 0954.51001
1 2 Kaplansky, Irving (1972), "Fields: Cubic and quartic equations", Fields and Rings, Chicago Lectures in Mathematics (2nd ed.), University of Chicago Press, ISBN 0-226-42451-0, Zbl 1001.16500
↑ Rotman, Joseph (1998), "Galois groups of quadratics, cubics, and quartics", Galois Theory (2nd ed.), Springer-Verlag, ISBN 0-387-98541-7, Zbl 0924.12001
↑ van der Waerden, Bartel Leendert (1991), "The Galois theory: Equations of the second, third, and fourth degrees", Algebra , vol. 1 (7th ed.), Springer-Verlag, ISBN 0-387-97424-5, Zbl 0724.12001

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Tignol-1] 1 2 Tignol, Jean-Pierre (2016), "Quartic equations", Galois' Theory of algebraic equations (2nd ed.), World Scientific, ISBN 978-981-4704-69-4, Zbl 1333.12001

[Brookfield-2] 1 2 Brookfield, G. (2007), "Factoring quartic polynomials: A lost art" (PDF), Mathematics Magazine , 80 (1): 67–70, doi:10.1080/0025570X.2007.11953453, JSTOR 27642994, S2CID 53375377, Zbl 1227.97040, archived from the original (PDF) on 2015-02-21

[3] Hartshorne, Robin (1997), "Construction problems and field extensions: Cubic and quartic equations", Geometry: Euclid and Beyond, Springer-Verlag, ISBN 0-387-98650-2, Zbl 0954.51001

[Kaplanski-4] 1 2 Kaplansky, Irving (1972), "Fields: Cubic and quartic equations", Fields and Rings, Chicago Lectures in Mathematics (2nd ed.), University of Chicago Press, ISBN 0-226-42451-0, Zbl 1001.16500

[5] Rotman, Joseph (1998), "Galois groups of quadratics, cubics, and quartics", Galois Theory (2nd ed.), Springer-Verlag, ISBN 0-387-98541-7, Zbl 0924.12001

[6] van der Waerden, Bartel Leendert (1991), "The Galois theory: Equations of the second, third, and fourth degrees", Algebra , vol. 1 (7th ed.), Springer-Verlag, ISBN 0-387-97424-5, Zbl 0724.12001

[1]

[2]

[3]

[4]

[5]

[6]