Theory of Equations and Abstract Algebra — Study Notes (Polynomials, Roots, and Abstract Algebra)

Polynomials in One Variable

Definition 1.1.1 (Polynomial in x)
- An expression of the form $a0x^n + a1x^{n-1} + \cdots + a_n$ with real or complex coefficients, where $x$ is the variable, is called an integral rational function of x or a polynomial in x.
- Coefficients: $a0, a1, \dots, an.$ Leading term: if $a0 \neq 0$ , polynomial is of degree $n$ with leading term $a_0x^n.$
- Example: $f(x) = 3x^3 - x + 2, \quad g(x) = 4x^4 - x^3 + 2x - 1, \quad h(x) = \sqrt{2}x^2 - (3 + \sqrt{2})x + 4.$ f(-1)=0, g(i)=3+3i, h(1)=1.
1.1.1 Multiplication of Polynomials
- Standard operation: addition, subtraction, multiplication.
- Detached coefficients method (aka coefficient-by-coefficient multiplication) to avoid writing powers of x explicitly.
- Example: Multiply (x^2 - x + 1)(x^2 + x + 1) $using detached coefficients; result is$ x^4 + x^2 + 1.
- Remark: Leading term: if leading terms are a0x^n $and$ b0x^m $, product leading term is$ a0b0x^{n+m}; if product is zero, one factor must be identically zero.
1.1.2 Division of Polynomials
- Given f(x) = a0x^n + \dots,\; g(x) = b0x^m + \dots $with$ n \ge m $, division yields$ f(x) = g(x)q(x) + r(x) $with$ \deg r < m or r identically zero.
- Coefficients-deduction process mirrors the usual long division; kept in detached-coefficients form as well.
- Example: dividing x^8 + x^7 + 3x^4 - 1 $by$ x^4 - 3x^3 + 4x + 1 $yields quotient$ x^4 + 4x^3 + 12x^2 + 32x + 82 $and remainder$ 194x^3 - 140x^2 - 360x - 83.
- Remark: If remainder is 0, then the divisor is a factor: f(x) = g(x)q(x).
1.1.3 The Remainder Theorem
- The remainder of division of f(x) $by$ x-c $equals$ f(c).
- Proof: from f(x)=(x-c)q(x)+f(c) $, substitute$ x=c $to get$ f(c)=r=f(c).
- Examples: divisibility tests via remainder: e.g., f(x) $is divisible by$ x+3 $if and only if$ f(-3)=0.
1.1.4 Synthetic Division
- Quick method to obtain quotient and remainder when dividing by x-c.
- Outline: write coefficients of f(x) $as a top row; perform additions with a constant multiplier$ c $to generate the bottom line, which gives coefficients of the quotient, and the last entry gives the remainder$ r=f(c).
- Example: dividing 3x^6-7x^5+5x^4 - x^2 -6x -8 $by$ x+2 $yields quotient$ 3x^5-13x^4+31x^3-62x^2+123x-252 $and remainder$ 496.
1.1.5 Taylor’s Formula
- For a polynomial f(x) $with derivatives, expand around$ x=c:
- f(x) = f(c) + f'(c)(x-c) + \frac{f''(c)}{2!}(x-c)^2 + \dots + \frac{f^{(n)}(c)}{n!}(x-c)^n.$n
- Examples show computing coefficients via derivatives at the expansion point.

Algebraic Equations and Their Roots

1.2.1 Algebraic Equations (Definition)
- If f(x) $is a polynomial (real or complex coefficients) and we set$ f(x)=0, the solutions are roots. The degree of the equation equals the degree of the polynomial (n), unless the leading coefficient vanishes.
- If c is a root, then by the Remainder Theorem, f(x)=(x-c)f_1(x) $; conversely, if$ f(x) $is divisible by$ x-c $, then$ c is a root.
- If there are distinct roots c,c1,\dots,c{m-1} $, then$ f(x) $is divisible by$ (x-c)(x-c1)\dots(x-c{m-1}). A polynomial of degree n can have at most n distinct roots.
- If m roots are known, the depressed equation is f(x)/(x-c)(x-c1)\dots(x-c{m-1})=0 $of degree$ n-m.
Remarks on multiplicity and factorization
- Root multiplicity definition: if root a $occurs as the factor$ (x-a)^\alpha $, then root$ a $has multiplicity$ \alpha; simple root if multiplicity 1.
- When a is a root of multiplicity \alpha $, then$ f^{(k)}(a)=0 $for all$ k<\alpha $and$ f^{(\alpha)}(a)\neq 0.
1.2.2 Identity Theorem
- If two polynomials f(x) $and$ f1(x) $have equal values at more than the degree many distinct points, they are identical:$ f(x)\equiv f1(x).
1.2.3 Fundamental Theorem of Algebra
- Every polynomial of degree n $with complex coefficients has exactly$ n $roots in the complex plane (counted with multiplicity). Thus$ f(x) $factors into linear factors over the complex numbers:$ f(x)=a0\,(x-\alpha1)(x-\alpha2)\cdots(x-\alphan).
1.2.4 Imaginary Roots of Equations with Real Coefficients
- Complex roots of a real-coefficient polynomial occur in conjugate pairs. Nonreal roots come in pairs a\pm bi.
- Any real polynomial factors into real linear and quadratic factors.
- Example: roots of x^4+1=0 $are$ 1\pm i\sqrt{2}, -1\pm i\sqrt{2} $; factorization into real quadratics:$ x^4+1=(x^2-\sqrt{2}x+1)(x^2+\sqrt{2}x+1).
1.2.5 Relations between Roots and Coefficients
- For polynomial f(x)=a0x^n+a1x^{n-1}+\dots+an $with roots$ \alpha1,\dots,\alpha_n,
- f(x)=a0\prod{i=1}^n(x-\alpha_i).
- Elementary symmetric sums: define s1=\sum \alphai,\; s2=\sum{i<j}\alphai\alphaj,\; \dots,\; sn=\alpha1\alpha2\cdots\alphan. Then the relationships with coefficients are
- -\frac{a1}{a0}=s1,\quad \frac{a2}{a0}=s2,\quad \dots,\quad (-1)^n\frac{an}{a0}=s_n.
Examples illustrate extracting roots from these relations (x^3-6x^2+11x-6=0 gives roots 1,2,3, etc.).

Rational Roots and Limits of Roots

1.3.1 Limits of Roots
- Upper limit of positive roots: a positive number that exceeds all positive roots.
- Lower limit of negative roots: a negative number smaller than all negative roots.
- For complex-coefficient polynomials, an upper bound on the moduli of all roots can be defined using an auxiliary equation with transformed coefficients.
1.3.2 A Method to Find an Upper Limit of Positive Roots
- Use the synthetic-division chain f0=a0, f1=x f0 + a1, …, fn = x f{n-1} + an.
- If for some c>0 the numbers f1(c),…,f{n-1}(c) are nonnegative and f_n(c)>0, then c is an upper bound for positive roots.
- If during the process some fk(c) becomes negative, increase c and retry until all f1(c),…,fn(c) are nonnegative with fn(c)>0.
1.3.3 Limit for Moduli of Roots
- For a polynomial with complex coefficients, bound the moduli of roots by solving an auxiliary equation with coefficients equal to absolute values of the original coefficients:
- If |x|=R and Ai=|ai|, then an upper bound R is found from A0R^n - A1R^{n-1} - \dots - An = 0.
1.3.4 Integral Roots
- Rational root test: integral roots must divide the constant term an. Use depressed equations after synthetic division to reduce the search space.
- Exclude divisors using limits and divisibility rules (e.g., if c is a root, then f(a) is divisible by c-a).
1.3.5 Rational Roots
- If the polynomial has a rational root x=r/s in lowest terms, then y=a0 x with a0 the leading coefficient is an integer root of a monic polynomial; thus the rational root must actually be integral after clearing denominators. Practical method uses substitution x=y/k to achieve integrality; choose k to keep coefficients integral.

Cubic and Biquadratic Equations

1.4.1 What is the Solution of an Equation?
- For first-degree equations: x = -b/a.
- For quadratic: x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}.
- Solving a quadratic by reduction to a simpler form; extraction of square roots leads to a related quadratic.
1.4.2 Cardan’s Formulas (Cubic)
- General cubic: x^3+ax^2+bx+c=0. $Substitute$ x=y-rac{a}{3} to remove the quadratic term, yielding $y^3+py+q=0$ with
- p=b-rac{a^2}{3},\quad q=c-rac{ab}{3}+rac{2a^3}{27}.
- Then set y=u+v $with the constraint$ uv=-\frac{p}{3} $, so that$ u^3+v^3=-q. $Solve the quadratic in$ t^2+qt-\frac{p^3}{27}=0 $to obtain$ u^3,v^3 = -\frac{q}{2} \pm \sqrt{\Delta}/2 $where$ \Delta = 4p^3+27q^2.
- Then take cube roots with the usual cube-root choices and assemble roots with complex cube roots of unity. This yields Cardan’s formulas:
- y_1 = \sqrt[3]{A} + \sqrt[3]{B},
- with A,B such that A+B=-q and AB=-\frac{p^3}{27}.
1.4.3 Discussion of Solution
- When p and q are real, the discriminant $\Delta = 4p^3+27q^2$ decides the nature of roots: positive, zero, or negative. If \Delta>0, one real root and two complex; if $\Delta=0$ , multiple roots; if \Delta<0, casus irreducibilis: three real roots but Cardan’s formula uses cube roots of complex numbers.
1.4.4 Irreducible Case (Casus irreducibilis)
- For \Delta<0, cube roots involve complex numbers; yet all three roots are real. This case cannot be expressed by real radicals alone.
1.4.5 Trigonometric Solution
- For casus irreducibilis, alternate expressions via trigonometric forms: write roots in terms of cosines using $y_j = 2\sqrt{-\frac{p}{3}}\cos\left(\frac{\phi+2\pi(j-1)}{3}\right)$ where $\cos\phi = \frac{-q/2}{\sqrt{(-p/3)^3}}.$
1.4.6 Solution of Biquadratic Equations
- Ferrari’s method: solve general biquadratic $x^4+ax^3+bx^2+cx+d=0$ by reducing to a quadratic in x^2 via an auxiliary parameter y, leading to resolvent cubic
- The resolvent cubic: $y^3 + py + q = 0$ with certain p,q; once y is found, solve two quadratics to get the four roots.
- Example: $x^4+4x-1=0$ has resolvent cubic $y^3+4y-16=0$ with rational root y=2; then solve quadratics to obtain roots.

Separation of Roots

1.5 Separation of real roots
- A real root is isolated if an interval contains exactly one root; roots are separated if each lies in its own interval.
- Theorem/idea: count real roots in an interval via sign changes; Rolle’s theorem and Descartes’ Rule provide tools.
1.5.1 Sign of a polynomial for small/large x
- For small |x|, the sign is that of the constant term; for large |x|, the sign is that of the leading term if degree is odd, and sign behaviour depends on parity for large magnitudes.
1.5.2 Rolle’s Theorem
- Between any two consecutive real roots of f, there is at least one real root of f′. Consequently, between two consecutive real roots of f there is at least one root of f′ (and an odd number in total).
1.5.3 Corollaries (Descartes’ Rule of Signs and root-separation)
- Descartes’ Rule: number of positive real roots is at most the number of sign variations in the coefficient sequence, differing by an even number; similarly for negative roots via f(-x).
1.5.4 Examples and practice on sign analysis, root counting, and interval localization.
1.5.5 An Important Identity and Lemma
- For roots x1, …, xn of f, identity for derivatives: f′(x)/f(x) = sum{i=1}^n 1/(x-xi).
- This leads to relationships between local behaviour near roots and multiplicities; used to study root stability under perturbation.
1.5.6 Rolle’s Theorem (detailed usage)
- Consequences for number of real roots in intervals via sign changes of f′ and f.
1.5.7 Descartes’ Rule of Signs (detailed examples)
- Count sign variations to bound positive roots; similarly for f(-x) for negative roots.

Symmetric Functions

1.6.1 Definition of Symmetric Functions
- P(x1,…,xn) is symmetric if invariant under any permutation of the variables.
- Examples: $x1^2+x2^2+x3^2,\;x1^3+x2^3+x3^3-3x1x2x3,\;x1+x2+x3+x_4$ are symmetric.
1.6.2 Practical Methods
- Sigma (or S-) functions: sums of terms obtained by symmetrizing a monomial, used to express symmetric polynomials in terms of elementary symmetric polynomials.
- Elementary symmetric polynomials for n variables: $e0=1,\, e1=\sum xi,\, e2=\sum{i<j} xi xj,\, …, en= x1x2…x_n.$
1.6.3 Examples
- For three variables: $e1=x+y+z,\; e2=xy+xz+yz,\; e_3=xyz.$
- A symmetric polynomial can be expressed as a linear combination of sigma functions, or equivalently in terms of the elementary symmetric polynomials.
1.6.2 Practical Methods (continued)
- Example: compute representation of given symmetric product in terms of p,q,r,s (the elementary symmetric elements). This includes formulas for transforming sigma to elementary symmetric sums.
1.6.3 Example calculations
- Worked examples showing how to decompose a symmetric function into sums of powers and then into elementary symmetric polynomials (p, q, r, s notation).

Module 2: Abstract Algebra

2.1 Integer Modulo n
- Congruence class [a]_n = {x ∈ Z | x ≡ a (mod n)}; Zn is the set of all congruence classes modulo n.
- Operations well-defined: addition and multiplication on Zn are defined by representatives: $[a]n + [b]n = [a+b]n,\; [a]n\cdot[b]n=[ab]n.$
- Additive inverse: $-[a]n = [-a]n.$
- Examples illustrate equivalence classes, e.g., in Z12.
2.1.1 Properties of Equivalence Classes
- Associativity, commutativity, distributivity, identities, and additive inverses hold for congruence classes.
- Additive identity: $[0]n.$ Multiplicative identity: $[1]n.$
2.1.3 Divisors of Zero and Units
- A nonzero class [a]n is a divisor of zero if there exists nonzero [b]n with [a]n[b]n=[0]_n.
- A unit is a congruence class [a]n with a multiplicative inverse: exists [b]n such that [a]n [b]n = [1]_n, which happens iff gcd(a,n)=1.
2.1.4 Inverses in Zn
- If gcd(a,n)=1, then there exists an inverse [a]^{-1}_n; else not.
- Example: in Z8, [7]^{-1}8=[1]8, [3]^{-1}8=[3]8, but [2]8 has no inverse (it’s a zero divisor).
2.1.5 Properties of Inverses and Units
- If [a]_n is a unit, then it cannot be a divisor of zero (proved via contradiction).
2.1.6 Theorem and propositions on units and divisors
- Key results: (i) gcd(a,n)=1 is equivalent to [a]_n being a unit; (ii) every nonzero element is either a unit or a zero divisor.
2.2 Equivalence Relations
- Equivalence relation on a set S: reflexive, symmetric, transitive; notation ∼. Equivalence classes [a].
- Quotient set S/∼ is the set of equivalence classes (a partition of S).
- Example: rational numbers via pairs (m,n) with mq=np; defines the usual rational numbers as S/∼.
- Proposition: every element of S lies in exactly one equivalence class; partitions correspond to equivalence relations.
2.3 Permutations
- A permutation is a bijective map from a set onto itself; Sym(S) is the set of all permutations of S; Sn denotes permutations of {1,…,n}.
- Counting: number of elements in Sn is n!.
- Composition of permutations and inverse, cycles, transpositions, and product decompositions into disjoint cycles.
- Key facts: every permutation is product of disjoint cycles; order of a permutation is lcm of the lengths of its disjoint cycles.
- Parities: even/odd permutations; product parity preserved under decomposition; sign determined by parity of number of transpositions.
2.4 Groups
- A group is a set with a binary operation satisfying closure, associativity, identity, and inverses.
- Examples: (R,+) is a group; Sym(S) is a group under composition; GLn(R) is a group under matrix multiplication; (R×,×) similarly forms a group.
- Subgroups, cancellation, and fundamental properties.
- Abelian (commutative) groups defined; examples: (Z,+), Zn under addition.
- Finite groups and order |G|; cyclic groups generated by an element a with G=(a).
2.5 Subgroups
- Subgroup criteria and cyclic subgroups: (a) = {a^n | n ∈ Z} is a subgroup.
- Properties and inclusion: if a ∈ K for a subgroup K, then (a) ⊆ K.
- Examples: cyclic groups like Z6, Klein four-group Z2×Z2, Z2×Z5, etc.

Module 3: Abstract Algebra (Isomorphisms, Further Structures)

3.1 Constructing Examples
- How to build groups of order n; cyclic vs non-cyclic cases; common examples and tables.
3.2 Isomorphism
- Group isomorphism definition; properties of isomorphisms; composition/preservation of structure; inverse mappings.
- Examples: R with + and R+ with × are isomorphic via exponential/log maps; Z4 vs Z2×Z2 nonisomorphic; Z6 ≅ Z2×Z3.
- Theorem: isomorphism preserves order of elements, abelian-ness, and cyclic property.
3.3 Cyclic Groups
- Every subgroup of a cyclic group is cyclic.
- Structure of cyclic groups: infinite cyclic isomorphic to Z; finite cyclic of order n isomorphic to Zn.
- Theorems on orders, generators, and subgroup structure: subgroups are precisely the sets generated by an element a^k with k | n.
3.4 Permutation Groups and Symmetric/Alternating Groups
- Cayley’s theorem: every group is isomorphic to a subgroup of a symmetric group on itself.
- Alternating group An: even permutations; properties and examples.
- The sign of a permutation and its relation to the sign of ∆n (Vandermonde determinant) in terms of parity.
3.5 Homomorphism
- Group homomorphism definition; kernel ker(φ); image φ(G1); fundamental homomorphism theorem; first isomorphism theorem: G1/ker(φ) ≅ φ(G1).
- Normal subgroups and kernels; normality and quotient groups.
3.6 Automorphisms and Inner Automorphisms
- Aut(G) and Inn(G); inner automorphisms defined by ia(x)=axa^{-1}; Inn(G) is normal in Aut(G).
- Center Z(G) and relation G/Z(G) ≅ Inn(G).
3.7 Classical results and examples
- Examples include Aut(Z), Inn(Z), Aut(Z2×Z2) ≅ S3, etc.

Module 4: Cosets, Normal Subgroups, and Ring Theory

4.1 Cosets
- Left cosets aH and right cosets Ha; index [G:H] is the number of left cosets.
- Examples: cosets in Z12 with subgroup
4.2 Isomorphism Theorems
- First Isomorphism Theorem: for φ: G1 → G2 with K=ker(φ), (HN)/N ≅ H/(H∩N) in a suitable setting; often stated as G1/ker(φ) ≅ φ(G1).
- Second Isomorphism Theorem: If N ⊆ H ⊆ G with N normal in G, then (G/N)/(H/N) ≅ G/H.
- Third isomorphism-type results and the behavior of products of normal subgroups.
4.3 Commutative Rings; Integral Domains
- Definitions: rings with two operations (+, ·); distributive laws; ring axioms; unity; additive inverse; additive/multiplicative identities.
- Subrings: a subset that is a ring with the same identity; examples include Zn within Z; Z[i] in the Gaussian integers; polynomial rings R[x] as polynomials over a ring R.
- Examples: subrings of Zn and Zn-like structures; polynomial rings construction; ring of sequences T = {(a0,a1,…) with finitely many nonzero} with formal power series-like structure; T is a ring isomorphic to R[x] in spirit.
4.4 (Implied) Center, Automorphisms, and further ring-theoretic notions
- While not exhaustively listed here, the text covers rings, subrings, and related constructions through Module 4.

Key Formulas and Theorems (selected)

Taylor expansion around c: $f(x)=\sum_{k=0}^n \frac{f^{(k)}(c)}{k!}(x-c)^k.$
Remainder Theorem: remainder of division of f by (x-c) equals $f(c).$
Fundamental Theorem of Algebra: every nonconstant polynomial with complex coefficients has a root in C; hence $f(x)=a0\prod{i=1}^n (x-\alpha_i).$
Viète relations (between roots and coefficients): if roots are $\alphai$ , then for $f(x)=a0x^n+\cdots+an$ , we have $-a1/a0 = \sum \alphai,\ a2/a0 = \sum{i<j} \alphai\alphaj,\dots, (-1)^n an/a0 = \alpha1\alpha2\cdots\alphan.$
Cardan’s formulas (summary): to solve $x^3+ax^2+bx+c=0$ reduce to $y^3+py+q=0$ with $p=b-\frac{a^2}{3},\quad q=c-\frac{ab}{3}+\frac{2a^3}{27}$ and set $y=u+v$ with $uv=-\frac{p}{3},\; u^3+v^3=-q$ ; solve the resolvent, then express roots via cube roots and cube roots of unity (with possible casus irreducibilis when \Delta=4p^3+27q^2<0).
Descartes’ Rule of Signs: the number of positive real roots is at most the number of sign changes in the coefficient sequence, minus an even number; similarly for f(-x) to count negative roots.
Lagrange’s Theorem (finite groups): for a subgroup H of a finite group G, the order of H divides |G|.
Cayley’s Theorem: every group is isomorphic to a subgroup of a symmetric group.
Direct products: if G1 and G2 are groups, then G1×G2 is a group; the order of (a1,a2) is lcm(order(a1),order(a2)).
First Isomorphism Theorem (outline): if φ: G1 → G2 is a homomorphism with kernel K, then G1/K ≅ φ(G1).
Normal subgroups and quotient groups: if N ◁ G, then G/N is a group with coset multiplication; kernel of projection π is N.
Center and Inn(G): Z(G) is normal; G/Z(G) ≅ Inn(G).
Polynomial rings: R[x] is the ring of polynomials over R; elements are finite sums a_i x^i with coefficients in R.

// Quick reference on notation used in this material

Polynomial f(x) with degree n: leading coefficient a0, coefficients a1,…,a_n.
Roots: positions where f(x)=0; multiplicity alpha means (x-α)^α divides f(x).
Sigma/S-functions: sums over symmetric index selections; expressed in terms of elementary symmetric polynomials e_k.
Congruence modulo n: [a]_n denotes the residue class of a modulo n; Zn denotes the ring of residue classes.
Group concepts: G, H, N, etc.; isomorphisms ∼=; homomorphisms; kernels; images; quotient structures; cyclic groups; dihedral groups Dn; symmetric groups Sn; alternating groups An.

Notes and connections

The theory connects polynomial algebra (roots, factors, coefficients) to abstract algebra (groups, rings, fields) through structures like symmetric polynomials, factorization, and homomorphisms.
Remainder and Factor Theorems provide practical tools for factoring polynomials and solving equations, while Cardan’s method shows the historical depth and complexity of solving higher-degree equations.
The Descartes Rule, Rolle’s Theorem, and separation of roots give powerful qualitative tools for real-root analysis without explicit root finding.
In Abstract Algebra, the structure of groups, rings, and modules underpins many areas of mathematics; core ideas include homomorphisms, kernels, quotients, and the interplay between structure and symmetry (Cayley, automorphisms, centers).
The material emphasizes both computational techniques (division, synthetic division, rational root testing) and structural theory (isomorphisms, normal subgroups, and factor groups), illustrating how computational methods fit into a broader algebraic framework.