Real
and Complex Algebra| 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; Golan 04 [II/III].
Algebraic Equations > s.a. functions [polynomials];
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,
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, i not= j 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 
(b2
– 4ac)1/2]/2a .
* Cubic: Consider the equation x3 + ax2 + bx + c = 0; Setting x = y – a/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 + 
[a1 (a12 –
4a2 + 4y1)1/2] z +
[y1 
(y12 –
4a4)1/2]
= 0 ,
where y1 is
the real solution of y3 – a2 y2
+ (a1a3 –
4a4) y + (4a2a4 – a32
– a12a4)
= 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: Pesic 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; 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 +
(A2–B)1/2]
/ 2}1/2 {[A –
(A2–B)1/2]
/ 2}1/2 .
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send feedback and suggestions to bombelli at olemiss.edu – modified
16 oct 2008