Page 22 - Advanced Linear Algebra
P. 22
6 Advanced Linear Algebra
Theorem 0.6 A nonconstant polynomial ²%³ is irreducible if and only if it has
the property that whenever ²%³ ²%³ ²%³ , then either ²%³ ²%³ or
²%³ ²%³.
Theorem 0.7 Every nonconstant polynomial in -´%µ can be written as a product
of irreducible polynomials. Moreover, this expression is unique up to order of
the factors and multiplication by a scalar.
Functions
To set our notation, we should make a few comments about functions.
:
;
Definition Let ¢ : ¦ ; be a function from a set to a set .
)
1 The domain of is the set and the range of is .
;
:
)
2 The image of is the set im² ³ ~ ¸ ² ³ :¹ .
3 ) is injective (one-to-one ) , or an injection , if £ % & ¬ ² % ³ £ ² & . ³
4 ) is surjective (onto ) , or a surjection , if im ² ³ ~ . ;
;
)
5 is bijective , or a bijection , if it is both injective and surjective.
)
6 Assuming that ; , the support of is
supp² ³~¸ : ² ³£ ¹
If ¢ : ¦ ; is injective, then its inverse c ¢ im ² ³ ¦ : exists and is well-
defined as a function on im² ³ .
It will be convenient to apply to subsets of and . In particular, if ? :
:
;
and if @ ; , we set
²?³ ~ ¸ ²%³ % ?¹
and
c ² @ ³ ~ ¸ : ² ³ @ ¹
Note that the latter is defined even if is not injective.
(
Let ¢ : ¦ ; . If ( : , the restriction of to is the function O ¢ ( ¦ ;
(
defined by
O ² ³ ~ ² ³
(
for all ( . Clearly, the restriction of an injective map is injective.
<
In the other direction, if ¢ : ¦ ; and if : < , then an extension of to is
a function ¢ < ¦ ; for which O ~ .
:
