Real and Complex Algebra  

In General
* Idea: The branch of mathematics that studies number systems and operations within them.

Linear Algebra > s.a. matrices.
@ References: Gel'fand 89; Robinson 91 [+ 92 solutions to exercises]; Hsiung & Mao 98; Golan 04 [II/III]; Goodaire 13.

Algebraic Equations > s.a. functions [polynomials]; history of mathematics [the cubic and the Great Feud]; inequalities.
$ Fundamental theorem: The n-th order polynomial Q(z) = a0 + a1 z + ... + an zn has exactly n roots in the complex plane; Can be proved as a consequence of Cauchy's theorem; Since Q(z) has no poles,

\[ {1\over2\pi{\rm i}} \oint {Q'(z)\over Q(z)}\,{\rm d}z = {\rm number\ of\ zeroes\ of}\ Q\;,\]

and this is a continuous function of n of the coefficients of Q, e.g., (a0, a1, ..., an–1). Thus...
* Results: Formulae for third and fourth degree have been long known; In general, the roots of the equation xn + a1 xn–1 + ... + an = 0 satisfy the equalities ∑i xi = –a1, ∑ij xi xj = a2, ..., ∏i xi = (–1)n an, known as Viète formulas.
* Abel's theorem: No formula can be found to express the roots of a general equation of degree n > 4 in terms of arithmetic operations and radicals involving its coefficients (Ruffini; Abel; Galois used group theory – in fact his efforts with the quintic gave rise to the field).
* However: For the quintic, Hermite, Kronecker and Brioschi (XIX century) independently found solutions in terms of elliptic modular functions; Klein discovered a solution in terms of hypergeometric functions.
* Quadratic: The roots of the polynomial P(x) = ax2 + bx + c are given by

\[ x = {-b\pm\sqrt{\vphantom{\sqrt2}b^2-4ac}\over2a}\;.\]

* Cubic: Consider the equation x3 + ax2 + bx + c = 0; Setting x = ya/3, it becomes y3 + py + q = 0, with p = –a3/3 and q = 2a3/27 – ab/3 + c; This is simpler to solve (the solution was found in the XVI century), and gives

y = [–q/2 + (q2/4 + p3/27)1/2]1/3 + [–q/2 – (q2/4 + p3/27)1/2]1/3,

where the values of the cubic roots are chosen so that their product is –p/3 (there are 3 possibilities!).
* Quartic: The solutions of x4 + a1x3 + a2x2 + a3x + a4 = 0 are the four roots of

z2 + \(1\over2\)[a1 ± (a12 – 4a2 + 4y1)1/2] z + \(1\over2\)[y1 \(\mp\) (y12 – 4a4)1/2] = 0 ,

where y1 is the real solution of y3a2 y2 + (a1a3 – 4a4) y + (4a2a4a32a12a4) = 0.
* Modular: An algebraic equation relating f(x) and f(x2) or f(x3), ...; Solutions are called modular functions; Example: f(x) = 2 [f(x2)]1/2 / [1+f(x2)] (second-order); & Ramanujan.
@ General references: Tignol 01 [Galois' theory]; Pešić 03, Alekseev 04 [Abel's theorem]; Boswell & Glasser mp/05 [sextic, solvable].
@ Quintic: Livio 05 [and groups]; Glasser a0907 [solution of DeMoivre's equation]; Bârsan a0910 [applications of Passare-Tsikh solution].

Algebraic Functions
@ References: Artin 67.

Related Topics > s.a. Algebraic Geometry; Geometric Algebra; Hypercomplex Algebra; numbers [algebraic number]; series; summations.
* Square roots: For a complex number

(a + ib)1/2 = ± [ {[a + (a2+b2)1/2] / 2}1/2 + (–1)b < 0 i {[–a + (a2+b2)1/2] / 2}1/2] ;

A useful identity is

(A ± B1/2)1/2 ≡ {[A + (A2B)1/2] / 2}1/2 ± {[A – (A2B)1/2] / 2}1/2 .


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