Page 21 - Advanced Linear Algebra

Preliminaries 5

Polynomials
The set of all polynomials in the variable with coefficients from a field is
-
%
denoted by -´%µ . If ²%³  -´%µ , we say that ²%³ is a polynomial over . If
-
²%³ ~ b % b Ä b %

is a polynomial with £ , then is called the leading coefficient of ²%³

and the degree of ²%³ is , written deg ²%³ ~ . For convenience, the degree

of the zero polynomial is cB . A polynomial is monic if its leading coefficient
is .

Theorem 0.4 (Division algorithm ) Let ²%³Á ²%³  -´%µ where deg ²%³ € .
Then there exist unique polynomials ²%³Á ²%³  -´%µ for which
²%³ ~ ²%³ ²%³ b ²%³
where ²%³ ~ or  deg ²%³  deg ²%³ .

If ²%³ divides ²%³ , that is, if there exists a polynomial ²%³ for which

²%³ ~ ²%³ ²%³
then we write ²%³ “ ²%³ . A nonzero polynomial ²%³  -´%µ is said to split
over if ² % ³ can be written as a product of linear factors
-
²%³ ~ ²%c ³Ä²%c ³

where  - .

Theorem 0.5 Let ²%³Á ²%³  -´%µ . The greatest common divisor of ²%³ and
²%³, denoted by gcd ² ²%³Á ²%³³, is the unique monic polynomial ²%³ over -
for which
1) ²%³ “ ²%³ and ²%³ “ ²%³
)
2 if ²%³ “ ²%³ and ²%³ “ ²%³ then ²%³ “ ²%³ .
-
Furthermore, there exist polynomials ²%³ and ²%³ over for which
gcd² ²%³Á ²%³³ ~ ²%³ ²%³ b ²%³ ²%³
Definition The polynomials ²%³Á ²%³  -´%µ are relatively prime if
gcd² ²%³Á ²%³³ ~ . In particular, ²%³ and ²%³ are relatively prime if and
-
only if there exist polynomials ²%³ and ²%³ over for which
²%³ ²%³ b ²%³ ²%³ ~
Definition A nonconstant polynomial ²%³  -´%µ is irreducible if whenever
²%³ ~ ²%³ ²%³, then one of ²%³ and ²%³ must be constant.
The following two theorems support the view that irreducible polynomials
behave like prime numbers.