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*) = *a*_{0} + *a*_{1}* z* +
... + *a*_{n}* z*^{n} 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., (*a*_{0}, *a*_{1},
..., *a*_{n–1}). Thus...

* __Results__: Formulae for
third and fourth degree have been long known; In general, the roots of the equation
*x*^{n} + *a*_{1}
*x*^{n–1}
+ ... + *a*_{n} = 0 satisfy
the equalities ∑_{i }*x*_{i}
= –*a*_{1}, ∑_{i ≠ j}
*x*_{i }*x*_{j}
= *a*_{2}, ..., ∏_{i}
*x*_{i}
= (–1)^{n} *a*_{n},
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*) = *ax*^{2} + *bx* + *c* are given by

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

* __Cubic__: Consider the
equation *x*^{3} + *ax*^{2}
+ *bx* + *c* = 0; Setting *x* = *y* – *a*/3,
it becomes *y*^{3} + *py* + *q* =
0, with *p* = –*a*^{3}/3
and *q* = 2*a*^{3}/27 – *ab*/3 + *c*;
This is simpler to solve (the solution was found in the XVI century), and gives

*y* = [–*q*/2 + (*q*^{2}/4
+ *p*^{3}/27)^{1/2}]^{1/3} +
[–*q*/2 – (*q*^{2}/4
+ *p*^{3}/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
*x*^{4} + *a*_{1}*x*^{3} +
*a*_{2}*x*^{2} + *a*_{3}*x* +
*a*_{4} = 0 are the four roots of

*z*^{2} + \(1\over2\)[*a*_{1} ± (*a*_{1}^{2} –
4*a*_{2} + 4*y*_{1})^{1/2}] *z* +
\(1\over2\)[*y*_{1} \(\mp\) (*y*_{1}^{2} –
4*a*_{4})^{1/2}]
= 0 ,

where *y*_{1} is
the real solution of *y*^{3} – *a*_{2} *y*^{2}
+ (*a*_{1}*a*_{3} –
4*a*_{4}) *y* + (4*a*_{2}*a*_{4} – *a*_{3}^{2}
– *a*_{1}^{2}*a*_{4})
= 0.

* __Modular__: An algebraic
equation relating *f*(*x*) and *f*(*x*^{2})
or *f*(*x*^{3}),
...; Solutions are called modular functions; __Example__: *f*(*x*)
= 2 [*f*(*x*^{2})]^{1/2} /
[1+*f*(*x*^{2})] (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* + i*b*)^{1/2} = ± [
{[*a* + (*a*^{2}+*b*^{2})^{1/2}]
/ 2}^{1/2} +
(–1)^{b < 0} i {[–*a* +
(*a*^{2}+*b*^{2})^{1/2}]
/ 2}^{1/2}] ;

A useful identity is

(*A* ± *B*^{1/2})^{1/2} ≡ {[*A* +
(*A*^{2}–*B*)^{1/2}]
/ 2}^{1/2} ± {[*A* –
(*A*^{2}–*B*)^{1/2}]
/ 2}^{1/2} .

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22 jan 2016