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Polynomial Degrees and Definition of a Field

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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
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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:

2.  Closure of multiplication.
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.
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/
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Associated Topics:
College Definitions
College Imaginary/Complex Numbers
High School Basic Algebra
High School Definitions
High School Imaginary/Complex Numbers
High School Polynomials
High School Sets

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