2026-08-21
Idempotent hierarchy of algebraic equations. From second to fourth degree and back
From the second degree of equations to the fourth and back
In the papers "From Euler's Formula to Split Geometry" and "Multilevel Idempotent Splitting of Phase Planes," it was shown that a single algebraic element can have multiple independent projections. In this paper, we apply this principle to the roots of algebraic equations over complex numbers. Instead of a typical list of roots, we collect them in a single multilevel split element.
The basic idea is simple: each distinct root is associated with its own finite idempotent plane. Splitting the plane creates an additional independent value and thereby increases the possible degree of the minimal polynomial. If the values on two child planes coincide, these planes merge, and the degree of the minimal polynomial decreases.
The main principle of this article: the degree of a minimal polynomial is equal to the number of distinct values of an element on finite, mutually orthogonal, idempotent planes.
This article will present the general idempotent principle for solving algebraic equations of the second, third, and fourth degrees. We will begin by placing two roots of a quadratic equation on two original planes, then, through additional splitting, move on to three and four independent roots. After this, we will demonstrate the reverse transition from the fourth degree to the third and second by merging identical values on the child planes. In the final section of the article, the proposed approach will be illustrated with simple examples of solving a quadratic, a cubic, and two fourth-degree equations.
1. The First Level of Idempotent Splitting
Consider two mutually orthogonal idempotents \(\ep\) and \(\em\):
\[\tag{1} \ep^2=\ep,\qquad \em^2=\em,\qquad \ep\em=0,\qquad \ep+\em=1. \] Element
\[\tag{2} X=\ep u+\em v \] takes the value \(u\) on the \(\ep\) plane and the value \(v\) on the \(\em\) plane. Due to orthogonality, mixed products vanish. For example,
\[\tag{3} X^2=\ep u^2+\em v^2, \qquad X^n=\ep u^n+\em v^n. \] Therefore, for any polynomial \(F\), the componentwise identity holds.
\[\tag{4} \boxed{ F(X)=\ep F(u)+\em F(v) }. \] This identity is the algebraic basis for the entire further construction. The polynomial acts independently on each idempotent plane.
2. Quadratic Equation as the First Level
Let \(u\) and \(v\) be two different numbers. Let's construct a polynomial:
\[\tag{5} P_2(x)=(x-u)(x-v) =x^2-(u+v)x+uv. \] Substitute the element \(X_2=\ep u+\em v\) into it. On the first plane, the factor \(X_2-u\) vanishes, and on the second, the factor \(X_2-v\) vanishes. Formally, this follows from (4):
\[\tag{6} P_2(X_2) =\ep P_2(u)+\em P_2(v)=0. \] Therefore, both roots of the quadratic equation are combined into a single element:
\[\tag{7} \boxed{ X_2=\ep u+\em v }. \] This does not imply addition of the roots in the usual sense. Idempotents preserve their independence: projection onto \(\ep\) returns \(u\), and projection onto \(\em\) returns \(v\).
3. Two Types of Normalization
To avoid limiting the method to roots of the unit modulus, it is necessary to distinguish between algebraic and phase normalization.
First, an arbitrary equation
\[\tag{8} a_nx^n+a_{n-1}x^{n-1}+\ldots+a_0=0, \qquad a_n\ne0, \] by dividing by \(a_n\), it is reduced to reduced, or monic, form.
\[\tag{9} x^n+A_{n-1}x^{n-1}+\ldots+A_0=0. \] This normalization is always possible and does not impose any restrictions on the roots. After this, any nonzero root can be represented in polar form.
\[\tag{10} r_k=\rho_k e^{ig_k}, \qquad \rho_k=|r_k|. \] Pure phase notation \(r_k=e^{ig_k}\) is a special case of \(\rho_k=1\). The idempotent principle itself does not require unit moduli: the amplitudes \(\rho_k\) can be stored on the corresponding planes.
4. Phase Normalization of a Quadratic Equation
Consider the quadratic equation
\[\tag{11} x^2+Ax+B=0, \qquad B\ne0. \] Let \(x=\sqrt{B}\,z\). After dividing by \(B\), we get
\[\tag{12} z^2+\frac{A}{\sqrt{B}}z+1=0. \] If \(A,B\) are real, \(B>0\) and \(A^2<4B\), then the roots of equation (12) form a conjugate pair of unit modulus:
\[\tag{13} z_{1,2}=e^{\pm ig}, \qquad g=\arccos\left(-\frac{A}{2\sqrt{B}}\right). \] Returning to the original variable, we get
\[\tag{14} x_{1,2}=\sqrt{B}\,e^{\pm ig}.\] Both roots are combined in the element
\[\tag{15} \boxed{ X_2=\sqrt{B} \left( \ep e^{ig}+\em e^{-ig} \right) }. \] Thus, normalization does not remove the amplitude of the roots, but separates it from the phase part. In what follows, we will first construct common elements with arbitrary values, and then consider the most illustrative phase cases.
5. Second level of splitting
We introduce a second pair of idempotents \(\pmp\) and \(\pme\):
\[\tag{16} \pmp^2=\pmp,\qquad \pme^2=\pme,\qquad \pmp\pme=0,\qquad \pmp+\pme=1. \] It is assumed that the idempotents of the two levels commute. Then each primary plane can be split into two child planes:
\[\tag{17} \ep=\ep\pmp+\ep\pme, \qquad \em=\em\pmp+\em\pme. \] A complete two-level splitting yields four finite planes:
\[\tag{18} E_1=\ep\pmp,\qquad E_2=\ep\pme,\qquad E_3=\em\pmp,\qquad E_4=\em\pme. \] They are mutually orthogonal and add up to unity:
\[\tag{19} E_kE_l=0\quad(k\ne l), \qquad E_1+E_2+E_3+E_4=1. \] The four finite planes are not the individual \(\ep,\em,\pmp,\pme\), but their products \(\ep\pmp,\ep\pme,\em\pmp,\em\pme\).
6. Transition from Second to Third Degree
Let's start with a two-component element
\[\tag{20} X_2=\ep u+\em v. \] We split only the plane \(\em\) and allow its child components to take on different values \(v_1\) and \(v_2\):
\[\tag{21} \em v \longrightarrow \em\left(\pmp v_1+\pme v_2\right). \] We obtain an element with three independent end values:
\[\tag{22} \boxed{ X_3=\ep u+ \em\left(\pmp v_1+\pme v_2\right) }. \] It corresponds to the polynomial
\[\tag{23} P_3(x)=(x-u)(x-v_1)(x-v_2). \] Expanding the brackets, we obtain
\[\tag{24} \begin{aligned} P_3(x)={}&x^3-(u+v_1+v_2)x^2\\ &+(uv_1+uv_2+v_1v_2)x-uv_1v_2. \end{aligned} \] Substitution (22) makes this polynomial vanish independently on three planes: \(\ep\), \(\em\pmp\), and \(\em\pme\). Thus, splitting one of the two original planes increases the possible degree of the minimal polynomial from two to three.
7. Phase Form of a Cubic Element
In the general amplitude-phase form, the three roots are written as
\[\tag{25} \boxed{ X_3= \ep\rho_1e^{ig} +\em\left( \pmp\rho_2e^{ih} +\pme\rho_3e^{il} \right) }. \] A particularly simple real case arises when one root is \(s=\pm1\), and the other two form a conjugate pair:
\[\tag{26} u=s,\qquad v_1=e^{ig},\qquad v_2=e^{-ig}. \] Then the unified solution takes the form
\[\tag{27} X_3=s\ep+ \em\left( \pmp e^{ig}+\pme e^{-ig} \right), \] and the cubic polynomial is expanded as follows:
\[\tag{28} P_3(x)=(x-s) \left(x^2-2x\cos g+1\right). \] After expanding the parentheses:
\[\tag{29} P_3(x)= x^3-(s+2\cos g)x^2 +(1+2s\cos g)x-s. \] Compare (29) with the normalized equation
\[\tag{30} x^3+Ax^2+Bx+C=0. \] For the phase class under consideration, the coefficients satisfy the conditions
\[\tag{31} C=-s,\qquad B=AC,\qquad C=\pm1. \] The phase of the conjugate pair is found directly from the coefficients:
\[\tag{32} \boxed{ g=\arccos\left(\frac{C-A}{2}\right) }. \] This is an explicit solution to a special phase-normalized class of cubic equations. For an arbitrary cubic equation, the amplitudes and phases of the three roots no longer have to have the form (26).
8. Transition from the third to the fourth power
To obtain the fourth independent value, we split the remaining plane \(\ep\):
\[\tag{33} \ep u \longrightarrow \ep\left(\pmp u_1+\pme u_2\right). \] A completely split element has the form
\[\tag{34} \boxed{ \begin{aligned} X_4={}&\ep\left( \pmp u_1+\pme u_2 \right)\\ &+\em\left( \pmp v_1+\pme v_2 \right). \end{aligned} } \] Its four values are located, respectively, on the planes \(\ep\pmp\), \(\ep\pme\), \(\em\pmp\), and \(\em\pme\). The minimal polynomial for various \(u_1,u_2,v_1,v_2) is equal to
\[\tag{35} \boxed{ P_4(x)= (x-u_1)(x-u_2)(x-v_1)(x-v_2) }. \] On every finite plane, one of the four factors vanishes, so \(P_4(X_4)=0\).
9. The General Split Element Theorem
The result obtained is not limited to degrees from two to four. Let \(n\) mutually orthogonal idempotents be given:
\[\tag{36} E_k^2=E_k,\qquad E_kE_l=0\(k\ne l),\qquad \sum_{k=1}^{n}E_k=1, \] and the element
\[\tag{37} X=\sum_{k=1}^{n}E_kr_k. \] Then for any polynomial \(F\), we have
\[\tag{38} \boxed{ F(X)=\sum_{k=1}^{n}E_kF(r_k) }. \] If the values \(r_1,\ldots,r_n\) are distinct, then the minimal polynomial of an element over the field of complex numbers is
\[\tag{39} \boxed{ m_X(x)=\prod_{k=1}^{n}(x-r_k) }. \] If some values coincide, the corresponding linear factors do not repeat in the minimal polynomial. This implies the central assertion:
\[\tag{40} \boxed{ \deg m_X= \text{number of distinct values of }X \text{ on finite planes} }. \] Thus, it is the number of distinct projection values, and not the formal number of introduced symbols, that determines the degree of the minimal polynomial.
10. Phase-amplitude form of a quartic equation
For a real equation, it is convenient to consider two complex conjugate pairs of roots:
\[\tag{41} u_{1,2}=\rho e^{\pm ig}, \qquad v_{1,2}=\sigma e^{\pm ih}. \] They are combined in the element
\[\tag{42} \boxed{ \begin{aligned} X_4={}&\rho\ep\left( \pmp e^{ig}+\pme e^{-ig} \right)\\ &+\sigma\em\left( \pmp e^{ih}+\pme e^{-ih} \right). \end{aligned} } \] Each conjugate pair forms a real square factor. Therefore, a fourth-degree polynomial takes the form
\[\tag{43} \boxed{ P_4(x)= \left(x^2-2\rho x\cos g+\rho^2\right) \left(x^2-2\sigma x\cos h+\sigma^2\right) }. \] Expanding the brackets, we get
\[\tag{44} \begin{aligned} P_4(x)={}&x^4 -2(\rho\cos g+\sigma\cos h)x^3\\ &+\left( \rho^2+\sigma^2 +4\rho\sigma\cos g\cos h \right)x^2\\ &-2\rho\sigma \left( \sigma\cos g+\rho\cos h \right)x +\rho^2\sigma^2. \end{aligned} \] Formula (43) shows the geometric meaning of the class of quartic equations under consideration: it is a union of two phase-amplitude quadratic equations on two primary planes. The second level of splitting separates the two roots within each pair.
11. Unit Phase Normalization of the Quantum Degree
Set \(\rho=\sigma=1\). Then, all four roots lie on the unit circle, and (43) simplifies:
\[\tag{45} P_4(x)= \left(x^2-2x\cos g+1\right) \left(x^2-2x\cos h+1\right). \] After multiplication:
\[\tag{46} \begin{aligned} P_4(x)={}&x^4 -2(\cos g+\cos h)x^3\\ &+(2+4\cos g\cos h)x^2\\ &-2(\cos g+\cos h)x+1. \end{aligned} \] Compare this formula with the equation
\[\tag{47} x^4+Ax^3+Bx^2+Cx+D=0. \] The purely phase real class is characterized by the conditions
\[\tag{48} \boxed{ D=1,\qquad C=A }. \] The cosines of the two phases satisfy the system
\[\tag{49} \cos g+\cos h=-\frac{A}{2}, \qquad \cos g\cos h=\frac{B-2}{4}. \] Therefore, the quantities \(t_1=\cos g\) and \(t_2=\cos h\) are the roots of the quadratic equation.
\[\tag{50} t^2+\frac{A}{2}t+\frac{B-2}{4}=0. \] Hence
\[\tag{51} \boxed{ \cos g= \frac{-A+\sqrt{A^2-4B+8}}{4}, \qquad \cos h= \frac{-A-\sqrt{A^2-4B+8}}{4} }. \] The desired phases are equal to
\[\tag{52} \boxed{ g=\arccos\left( \frac{-A+\sqrt{A^2-4B+8}}{4} \right), \quad h=\arccos\left( \frac{-A-\sqrt{A^2-4B+8}}{4} \right) }. \] For real phases, both quantities on the right-hand sides of (51) must belong to the interval \([-1,1]\). In this special class, the solution of a quartic equation is reduced to a quadratic equation in the cosines of the two phases.
12. Inverse transition: from the fourth power to the third
Let's return to the general four-component element (34) and set
\[\tag{53} u_1=u_2=u. \] The two child components of the plane \(\ep\) take the same value and merge:
\[\tag{54} \ep\left(\pmp u+\pme u\right) =\ep(\pmp+\pme)u =\ep u. \] Therefore,
\[\tag{55} X_4\longrightarrow X_3=\ep u+ \em\left(\pmp v_1+\pme v_2\right). \] If we retain the formal product of four factors, it takes the form
\[\tag{56} P_4(x)=(x-u)^2(x-v_1)(x-v_2). \] However, the element has only three distinct values. Therefore, its minimal polynomial does not contain a repeating factor:
\[\tag{57} \boxed{ m_{X_3}(x)=(x-u)(x-v_1)(x-v_2) }. \] So, the transition \(4\to3\) occurs not through the disappearance of the coefficient of \(x^4\), but through the coincidence of two projection values and the transition to the minimal polynomial.
13. Reverse transition: from thirdth power to the second
Now let
\[\tag{58} v_1=v_2=v. \] The child planes inside \(\em\) also merge:
\[\tag{59} \em\left(\pmp v+\pme v\right) =\em v. \] As a result
\[\tag{60} X_3\longrightarrow X_2=\ep u+\em v. \] The formal polynomial \((x-u)(x-v)^2\) is of degree three, but the minimal polynomial of an element is already square:
\[\tag{61} \boxed{ m_{X_2}(x)=(x-u)(x-v) }. \] 14. A unified hierarchy of increasing and decreasing degrees
All transitions can be collected into a single sequence:
\[\tag{62} \boxed{ \begin{aligned} X_2&=\ep u+\em v, \\[2mm] X_3&=\ep u+ \em\left(\pmp v_1+\pme v_2\right), \\[2mm] X_4&=\ep\left(\pmp u_1+\pme u_2\right) +\em\left(\pmp v_1+\pme v_2\right). \end{aligned} } \] When moving upward, one plane splits, and its single value is replaced by two independent ones. When moving down, two child values are equated, after which the corresponding idempotents are added back to the parent plane.
\[\tag{63} \boxed{ 2\xrightarrow{\text{split}}3 \xrightarrow{\text{split}}4, \qquad 4\xrightarrow{u_1=u_2}3 \xrightarrow{v_1=v_2}2 }. \] Splitting increases the number of possible distinct values of an element. Merging identical values decreases the degree of its minimal polynomial.
15. Example 1. A pure phase equation of the fourth degree
Let's solve the equation
\[\tag{64} x^4+x^2+1=0. \] It belongs to the phase-normalized class (47): \(A=C=0\), \(B=1\), \(D=1\). According to formula (51)
\[\tag{65} \cos g=\frac{\sqrt{4}}{4}=\frac12, \qquad \cos h=-\frac{\sqrt{4}}{4}=-\frac12. \] We choose
\[\tag{66} g=\frac{\pi}{3}, \qquad h=\frac{2\pi}{3}. \] The polynomial is factored into two phase square factors:
\[\tag{67} x^4+x^2+1 =\left(x^2-x+1\right) \left(x^2+x+1\right). \] The four roots are equal to
\[\tag{68} x_{1,2}=e^{\pm i\pi/3}, \qquad x_{3,4}=e^{\pm i2\pi/3}. \] The single split solution is:
\[\tag{69} \boxed{ \begin{aligned} X_4={}&\ep\left( \pmp e^{i\pi/3} +\pme e^{-i\pi/3} \right)\\ &+\em\left( \pmp e^{i2\pi/3} +\pme e^{-i2\pi/3} \right). \end{aligned} } \] Each of the four projections of element (69) is a root of equation (64), therefore \(P_4(X_4)=0\).
16. Example 2. A Quartic Equation with Different Amplitudes
Now consider an equation whose roots lie on circles of different radii:
\[\tag{70} x^4-2x^3+13x^2-18x+36=0. \] The polynomial is expanded as follows:
\[\tag{71} x^4-2x^3+13x^2-18x+36 =\left(x^2-2x+4\right) \left(x^2+9\right). \] Compare the first factor with the general phase-amplitude form \(x^2-2\rho x\cos g+\rho^2\). We get
\[\tag{72} \rho=2, \qquad \cos g=\frac12, \qquad g=\frac{\pi}{3}. \] For the second factor, we have
\[\tag{73} \sigma=3, \qquad \cos h=0, \qquad h=\frac{\pi}{2}. \] Therefore, the roots of the equation are
\[\tag{74} x_{1,2}=2e^{\pm i\pi/3}=1\pm i\sqrt3, \qquad x_{3,4}=3e^{\pm i\pi/2}=\pm3i. \] Let's combine them in one element:
\[\tag{75} \boxed{ \begin{aligned} X_4={}&2\ep\left( \pmp e^{i\pi/3} +\pme e^{-i\pi/3} \right)\\ &+3\em\left( \pmp e^{i\pi/2} +\pme e^{-i\pi/2} \right). \end{aligned} } \] The conjugate pair with modulus \(2\) is located on the \(\ep\pmp\) and \(\ep\pme\) planes, and the pair with modulus \(3\) is located on the \(\em\pmp\) and \(\em\pme\) planes. This example shows that the unit phase normalization is convenient, but not required.
17. An example of solving a third-degree equation
Consider the equation
\[\tag{76} x^3-2x^2+2x-1=0. \] Here \(A=-2\), \(B=2\), \(C=-1\). Conditions (31) are satisfied:
\[\tag{77} C=-1, \qquad B=AC=(-2)(-1)=2. \] From \(C=-s\), we find \(s=1\). The phase of the conjugate pair is given by formula (32):
\[\tag{78} g=\arccos\left( \frac{-1-(-2)}{2} \right) =\arccos\frac12 =\frac{\pi}{3}. \] Therefore,
\[\tag{79} x^3-2x^2+2x-1 =(x-1)(x^2-x+1), \] and the three roots are equal
\[\tag{80} x_1=1, \qquad x_{2,3}=e^{\pm i\pi/3}. \] Single split solution:
\[\tag{81} \boxed{ X_3=\ep+ \em\left( \pmp e^{i\pi/3} +\pme e^{-i\pi/3} \right) }. \] Plane \(\ep\) contains the root \(1\), and the child planes \(\em\pmp\) and \(\em\pme\) contain the conjugate roots.
18. The Final Example of a Quadratic Equation
In the previous examples, the quadratic factor appeared twice.
\[\tag{82} x^2-x+1=0. \] Its roots are equal to \(e^{\pm i\pi/3}\), so the split solution has the original two-component form:
\[\tag{83} \boxed{ X_2=\ep e^{i\pi/3} +\em e^{-i\pi/3} }. \] Thus, the same conjugate pair appears at the second, third, and fourth degree levels. It's not the pair itself that changes, but the number of additional independent values added through further splitting.
19. What exactly does the proposed method provide?
The idempotent construction combines all the roots of a polynomial into a single element, preserves their independence, and explains the relationship between the number of finite planes and the degree of the minimal polynomial. For phase-normalized classes, the coefficients directly determine the phases, as in formulas (32) and (52).
However, the decomposition itself
\[\tag{84} X=\sum_kE_kr_k \] is not a separate general formula for finding the unknowns \(r_k\) for an arbitrary polynomial. If the amplitudes and phases are not pre-related by the symmetry of the coefficients, their determination remains the original algebraic problem. The new content of the method lies in the geometry of the roots, the hierarchy of planes, and the natural mechanism for raising and lowering the degree.
For special normed families, the method simultaneously provides both a representation and a simple solution. For an arbitrary polynomial, it primarily yields a single spectrally idempotent form of all roots.
20. Generalization to Degree n
After \(m\) complete binary splitting levels, \(2^m\) finite orthogonal planes arise. Therefore, to accommodate \(n\) independent roots, it suffices to
\[\tag{85} \boxed{ m=\left\lceil\log_2 n\right\rceil }. \] For example,
\[\tag{86} n=2\Rightarrow m=1, \qquad n=3,4\Rightarrow m=2, \qquad n=5,6,7,8\Rightarrow m=3. \] It is not necessary to split all branches to the same depth. For three roots, one primary plane remains intact, and the second is divided into two. This is why two levels yield not only four but also three intermediate independent finite planes.
21. Conclusions
Two primary idempotents allow two roots of a quadratic equation to be combined. Splitting one primary plane creates three independent components and allows for a cubic minimal polynomial. Splitting both primary planes creates four components and a fourth-degree polynomial.
The inverse process also has a simple geometric meaning. If the values on two child planes are the same, they are added to the parent plane. The repeating factor is preserved in the formal polynomial of the higher degree, but disappears from the minimal polynomial of the element.
\[\tag{87} \boxed{ \text{splitting planes} \Longleftrightarrow \text{increasing the number of independent roots} } \] \[\tag{88} \boxed{ \text{merging identical values} \Longleftrightarrow \text{lowering the degree of a minimal polynomial} } \] Thus, the algebraic degree receives a clear geometric interpretation: it expresses the number of different values that a single multi-level split element takes on its finite idempotent planes.