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;
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,
∑i ≠ 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
\[ x = {-b\pm\sqrt{\vphantom{\sqrt2}b^2-4ac}\over2a}\;.\]
* 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 + \(1\over2\)[a1 ± (a12 − 4a2 + 4y1)1/2] z + \(1\over2\)[y1 \(\mp\) (y12 − 4a4)1/2] = 0 ,
where y1 is
the real solution of 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:
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 + (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 22 jan 2016