Polynomial Degrees and Definition of a Field

Date: 03/02/98 at 11:51:19
From: Seb Metelli
Subject: rings, fields, vector spaces...

Dear Dr. Math: 

I'm a student in a European (read: run by Brussels) school in northern 
Italy. At the moment in advanced maths, we're studying groups of 
different sorts, combining them with polynomials, complex numbers, 
vectors, matrices, etc.

My questions are: when you add 2 polynomials with i) rational, ii) 
complex, or iii) real solutions, do you get polynomials of the same 
degree with real/complex/real roots? If so, how does one prove this 
for all values?

N.B.: the polynomials would be members of a set containing all 
polynomials with rational/real/complex solutions.

Finally, I was wondering whether you could give me the definition of a 
field (as in commutativity, inverse, etc.).

Thank you,  

Seb Metelli

Date: 03/02/98 at 12:59:27
From: Doctor Rob
Subject: Re: rings, fields, vector spaces...

In short, the answer is no. A trivial counter-example is the pair of 
polynomials x + 1 and -x. Each has rational (hence real, and 
therefore complex) roots, but their sum is 1, which has no roots 
at all. It also has a different degree than the original polynomials.

If you insist that the summands and the sum have the same degree, the
answer is no for rational and real, but yes for complex roots. You can 
easily find counter-examples: x^2 - 1 and x^2 - 4 have rational roots, 
but their sum has irrational ones, and x^2 - 1 and -2*x^2 + 1/2 both 
have real (even rational!) roots, but their sum has imaginary ones. 
The only thing that saves the situation for complex roots is that 
every polynomial of degree n > 0 with complex coefficients has n 
complex roots. This is the Fundamental Theorem of Algebra.

A field is a commutative ring with 1 in which every nonzero element 
has a multiplicative inverse.

More basically, it is a set with two operations, addition and
multiplication, that satisfies the following axioms:

 1.  Closure of addition.
 2.  Closure of multiplication.
 3.  Associative Law of Addition.
 4.  Associative Law of Multiplication.
 5.  Distributive Law.
 6.  Existence of 0.
 7.  Existence of 1.
 8.  Existence of additive inverses (negatives).
 9.  Existence of multiplicative inverses (reciprocals), except for 0.
 10. Commutative Law of Addition.
 11. Commutative Law of Multiplication.

Examples of fields are Q (rationals), R (reals), C (complexes), and Z/
pZ (integers modulo a prime p).

-Doctor Rob, The Math Forum
Check out our web site http://mathforum.org/dr.math/