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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send feedback and suggestions to bombelli at olemiss.edu – modified 22 jan 2016