Agoston M.K. Computer Graphics and Geometric Modelling: Mathematics

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B.8.10. Theorem. Let k be a ﬁeld. A nonconstant polynomial f(X) Œ k[X] has a mul-

tiple factor g(X) if and only if g(X) is also a factor of f¢(X).

Proof. Let f(X) = g(X)

h(X). The product rule of the derivative easily leads to the

result.

B.8.11. Theorem. If k is a ﬁeld, then k[X] is a PID (and hence a UFD).

Proof. See [Mill58].

Only the polynomial ring in one variable over a ﬁeld is a PID. To see that

k[X

,...,X

] is not a PID if n > 1, one simply needs to convince oneself that the

ideal in k[X,Y] consisting of all polynomials whose constant term is 0 is not a prin-

cipal ideal.

Deﬁnition. Let f

, f

,..., f

Œ k[X]. A greatest common divisor of the f

, denoted by

gcd(f

,...,f

), is a largest degree polynomial g Œ k[X] that divides each of the f

. A

least common multiple of the f

, denoted by lcm(f

,...,f

), is a smallest degree poly-

nomial h Œ k[X] with the property that each f

divides h.

B.8.12. Theorem. Let f

, f

,..., f

Œ k[X].

(1) The polynomials gcd(f

,...,f

) and lcm(f

,...,f

) are unique up to scalar

multiple.

(2) <f

,...,f

> = <gcd(f

,...,f

)>.

Proof. This is easy. The key fact is that k[X] is a UFD.

B.8.13. Theorem. If k is a ﬁeld, then both k[[X

,...,X

]] and k[X

,...,X

]

are Noetherian.

Proof. First note that k satisﬁes the ascending chain condition since it has only two

ideals, namely, 0 and itself. Now use Corollary B.7.10 and Theorem B7.11.

Deﬁnition. Let K be a ﬁeld. If k is a subﬁeld of K, then K is called an extension (ﬁeld)

of k.

Deﬁnition. Let K

and K

be extension ﬁelds of a ﬁeld k. An isomorphism

s:K

Æ K

is called an isomorphism over k if s is the identity on k. If K = K

= K

then s is called an automorphism of K over k.

Deﬁnition. Let k be a subﬁeld of a ﬁeld K. If A Õ K, then k(A) will denote the small-

est subﬁeld of K containing k and A, more precisely,

k A F F is a subfield of K which contains k and A

()

=«

{}

B.8 Fields 849

If A consists of a single element a, then we shall usually write k(a) instead of k(A).

It is easy to show that the intersection of ﬁelds is a ﬁeld and so k(A) is a well-

deﬁned subﬁeld.

Deﬁnition. If K is an extension of a ﬁeld k and if K = k(a) for some element a, then

K is called a simple extension of k and the element a is called a primitive element of

K over k.

Deﬁnition. If k is a ﬁeld, then k(X) will denote the ﬁeld of quotients, or quotient

ﬁeld, of k[X]. More generally, k(X

,...,X

) will denote the ﬁeld of quotients of

k[X

,...,X

There are natural inclusions

B.8.14. Theorem. Let k be a subﬁeld of a ﬁeld K. If a Œ K, then either

or a is a root of an irreducible polynomial f(X) in k[X] and

Proof. See [Mill58]. Note that <f(X)> is a maximal ideal and so k[X]/<f(X)> is a ﬁeld

by Theorem B.8.5.

B.8.15. Theorem. Let K and K¢ be subﬁelds of ﬁelds E and E¢, respectively, and let

j:K Æ K¢ be an isomorphism. Let

and

Assume that p(X) is irreducible. If c Œ E and c¢ŒE¢ are roots of the polynomials p(X)

and p¢(X), respectively, then the isomorphism j extends to a unique isomorphism

j¢:K(c) Æ K¢(c¢) such that j¢(c) = c¢.

Proof. See [Dean66].

B.9 The Complex Numbers

This section will simply deﬁne the ﬁeld of complex numbers. Complex numbers play

an essential role in many areas of mathematics. Appendix E will discuss some of their

nontrivial properties, especially as they relate to analysis.

()

+◊◊◊+

()

Œ¢

[]

pX a a X a X KX

jj j

pX a a X a X KX

()

= + +◊◊◊+ Œ

[]

ka kX fX

()

[]

()

ka kX

()

kkX kXX kXX X

()

Ã◊◊◊Ã

()

112 12

, , ,..., .

850 Appendix B Basic Algebra

Identify (a,b) Œ R

with the formal expression a + ib. (0 + ib is abbreviated to

ib and i·1, to i.) Using this notation deﬁne arithmetic operations + and · on R

follows:

B.9.1. Theorem. (R

,+,·) is a ﬁeld denoted by C which contains R as a subﬁeld under

the identiﬁcation of a with (a,0).

Proof. Straightforward.

Thought of as ﬁeld elements, the elements of R

are called complex numbers. It is

easy to check that we have the well-known identity

in other words, -1 has a square root in C.

Deﬁnition. If z = a + ib Œ C, then a is called the real part of z and b is called the

imaginary part of z. Deﬁne functions Re(z) and Im(z) by

The complex conjugate of z, , and the modulus of z, |z|, are deﬁned by

and

Here are two simple facts that are easily checked:

(1) The modulus function is multiplicative, that is, |z

| = |z

||z

| .

(2) If z = a + ib, then we have the identity

B.10 Vector Spaces

Vector spaces are discussed in more detail in Appendix C. We only deﬁne them here

and list those basic properties that are needed in the section on ﬁeld extensions.

Deﬁnition. A vector space over a ﬁeld k is a triple (V,+,·) consisting of a set V of

objects called vectors together with two operations + and · called vector addition and

22 22

zab=+

za b=-i

Re Im .z a and z b

()

1=- ,

ab cd ac bd

a b c d ac bd ad bc

()

◊+

()

ii i

B.10 Vector Spaces 851

scalar multiplication, respectively, so that (V,+) is an abelian group, and for each a Œ

k and u Œ V, a·v Œ V. Furthermore, the following identities hold for each a, b Œ k and

u, v Œ V:

(1) (distributivity) a·(u + v) = a·u + a·v

(2) (distributivity) (a + b)·u = a·u + b·u

(3) (associativity) (ab)·u = a·(b·u)

(4) (identity) 1·u = u

For simplicity, the operator · is usually suppressed and one writes au instead of

a·u. Also, if the ﬁeld and the operations are clear from the context, the vector space

(V,+,·) will be referred to simply as the vector space V. It is easy to show that

and it is useful to deﬁne a subtraction operators for vectors by setting

N-dimensional Euclidean space R

, or n-space, is the most well-known example of

a vector space because it is more than just a set of points and admits a well-known

vector space structure over R. Namely, let u = (u

,...,u

), v = (v

,...,v

) Œ R

c Œ R, and deﬁne

One thinks of elements of R

as either “points” or “vectors,” depending on the context.

More generally, it is easy to see that if k is a ﬁeld, then k

is a vector space over k.

Function spaces are another important class of vector spaces. Let k be a ﬁeld and

let A be a subset of k

. Then the set of functions from A to k is a vector space over k

by pointwise addition and scalar multiplication. More precisely, if f, g : A Æ k and c Œ

k, deﬁne

Deﬁnition. A subspace of a vector space V is a subset of V with the property that it,

together with the induced operations from V, is itself a vector space.

For example, under the natural inclusions, the vector spaces R

, m £ n, are all

subspaces of R

f g x f x g x and cf x c f x+

()()

()

() ( )()

()()

f g cf k+Æ,:A

+= + +

()

uv uv

ccu cu

,...,

,..., .

uvu v-=+-

()

-=-

()

uu1

852 Appendix B Basic Algebra

Deﬁnition. Let V be a vector space over a ﬁeld k. Given a nonempty set of vectors

S = {u

,..., u

} in V, we deﬁne the span of the set, span(S), or the span of the vectors

in S, span(u

,...,u

), to be the set of all vectors that are linear combinations of these

vectors, that is,

We say that the set S spans X and the vectors u

,..., u

span a subspace X if

It is convenient to deﬁne span(f) = {0} .

It is easy to check that span(u

,...,u

) is a vector subspace of V.

Deﬁnition. Let V be a vector space over a ﬁeld k. A nonempty set of vectors

S = {u

,..., u

} in V, is said to be linearly dependent if

for some ﬁeld elements c

, c

,..., c

not all of which are zero. The set S and the

vectors u

, u

,..., u

are said to be linearly independent if they are not linearly depen-

dent. It is convenient to deﬁne the empty set to be a linearly independent set of vectors.

In other words, vectors are linearly independent if no nontrivial linear combina-

tion of them adds up to the zero vector. Two linearly dependent vectors are often called

collinear. Two nonzero vectors u

and u

are collinear if and only if they are multi-

ples of each other, that is, span(u

) = span(u

Deﬁnition. Let X be a subspace of a vector space. A set of vectors S is said to be a

basis for X if it is a linearly independent set that spans X.

Note. The deﬁnitions of linearly independent/dependent, span, and basis above dealt

only with ﬁnite collections of vectors to make the deﬁnitions clearer and will apply to

most of our vector spaces. On a few occasions we may have to deal with “inﬁnite dimen-

sional” vector spaces and it is therefore necessary to indicate what changes have to be

made to accommodate those. All one has to do is be a little more careful about what

constitutes a linear combination of vectors. Given an arbitrary, possibly inﬁnite, set of

vectors {v

}

aŒI

in a vector space, deﬁne a linear combination of those vectors to be a sum

where all but a ﬁnite number of the c

are zero. With this concept of linear combi-

nation, the deﬁnitions above and the next theorem will apply to all vector spaces.

B.10.1. Theorem. Every vector subspace of a vector space has a basis and the

number of vectors in a basis is uniquely determined by the subspace. Every set of lin-

early independent vectors in a vector space can be extended to a basis.

nn11

uu0+◊◊◊+ =

Xuu=

()

span S span

,..., .

span S span c c c k

nnni

()

= +◊◊◊+ Œ

{}

uu u u

111

,..., .

B.10 Vector Spaces 853

Proof. See [Dean66].

Theorem B.10.1 justiﬁes the following deﬁnition:

Deﬁnition. The dimension of a vector space V, denoted by dim V, is deﬁned to be

the number of vectors in a basis for it. A m-dimensional subspace of an n-dimensional

vector space is said to have codimension n–m.

Note. With our deﬁnitions above, the vector space that consists only of the zero

vector has dimension 0 and the empty set is a basis for it. This may sound a little

strange, but it gives us a nice uniform terminology without which some results would

get a little more complicated to state.

Deﬁnition. Let V and W be vector spaces over a ﬁeld k. A map T: V Æ W is called

a linear transformation if T satisﬁes

for all v Œ V, w Œ W, and a, b Œ k .

B.10.2. Theorem. The inverse of a linear transformation, if it exists, is a linear

transformation and so is the composite of linear transformations.

Proof. Easy.

Deﬁnition. A linear transformation is said to be nonsingular if it has an inverse;

otherwise, it is said to be singular. A nonsingular linear transformation is called a

vector space isomorphism.

Deﬁnition. Let V and W be vector spaces over a ﬁeld k. If T :V Æ W is a linear trans-

formation, then deﬁne the kernel of T, ker (T), and the image of T, im (T), by

and

B.10.3. Theorem. If T :V Æ W is a linear transformation between vector spaces V

and W, then the kernel and image of T are vector subspaces of V and W, respectively.

Furthermore,

Proof. Easy. See [John67].

Let W be a subspace of a vector space V over a ﬁeld k. It is easy to check that the

quotient group V/W becomes a vector space over k by deﬁning

dim dim dim ker .V =

()

im T T

()

[]

ÕvvV W.

ker T T

()

=Œ

()

{}

vV v 0

Ta b aT bTvw v w+

()

854 Appendix B Basic Algebra

for each coset v + W in V/W and c Œ k.

Deﬁnition. The vector space V/W is called the quotient vector space of V by W.

It is easy to show that the natural projection V Æ V/W that sends a vector in V

to the coset that it determines is a well-deﬁned surjective linear transformation whose

kernel is W.

B.10.4. Theorem. Let V and W be vector spaces over a ﬁeld k. Let v

, v

,..., v

a basis for V and let w

, w

,..., w

be any vectors in W. There exists a unique linear

transformation T:V Æ W such that T(v

) = w

, i = 1, 2,..., n.

Proof. Easy.

Let k be a ﬁeld and S a set. Deﬁne F(S) to be the set of “formal” sums

where r

Œ k and all but a ﬁnite number of the r

are 0. There is an obvious addition

and scalar multiplication making F(S) into a vector space. A more precise deﬁnition

of F(S) would deﬁne it to be the set of functions j:S Æ k that vanish on all but a ﬁnite

number of elements in S. We leave it to the reader to ﬁll in the technical details.

Deﬁnition. The vector space F(S) is called the free vector space over k with basis S.

B.10.5. Theorem. Let S be a set and V a vector space. If t:S Æ V is any map, then

there is a unique linear map T:F(S) Æ V with T(s) = t(s) for all s Œ S.

Proof. Easy. Like with free groups, the best way to think about this lifting of t to a

map T is via a commutative diagram:

ccvW vW+

()

B.11 Extension Fields 855

inclusion map i

F(S)

We have another universal factorization property and this is what actually deﬁnes F(S)

up to isomorphism.

B.11 Extension Fields

This section describes some fundamental properties of extension ﬁelds.

Deﬁnition. Let k be a subﬁeld of a ﬁeld K. An element a Œ K is said to be transcen-

dental over k if k(a) ª k(X). It is said to be algebraic over k if it is the root of a poly-

nomial f(X) Œ k[X]. If it is the root of an irreducible polynomial of degree n, then it

is said to be algebraic over k of degree n.

This deﬁnition of transcendental and algebraic agrees with the deﬁnition given for

rings in Section B.7.

B.11.1. Theorem. Let k be a subﬁeld of a ﬁeld K. If an element a Œ K is algebraic

over k, then a is the root of a unique irreducible polynomial with leading coefﬁcient

equal to 1 called the minimum polynomial of a over k.

Proof. See [Jaco64].

Deﬁnition. An extension K of a ﬁeld k is called an algebraic extension of k if every

element of K is algebraic over k. If K is not an algebraic extension of k, then it is called

a transcendental extension.

Deﬁnition. An extension K of a ﬁeld k is called a ﬁnite extension of k of degree n if K

is a ﬁnite dimensional vector space over k of dimension n. Otherwise, K is called an

inﬁnite extension of k of degree •. The degree of the extension is denoted by [K:k].

B.11.2. Theorem. If K is a ﬁnite extension of a ﬁeld k, then K is an algebraic exten-

sion of k.

Proof. See [Mill58].

B.11.3. Theorem. If K is a ﬁnite extension of a ﬁeld k, then K = k(q

,...,q

where the q

are algebraic over k.

Proof. See [Mill58].

B.11.4. Theorem. (Theorem of the Primitive Element) If k is a ﬁeld of characteris-

tic 0 and if q

, q

,..., andq

belong to some extension ﬁeld of k and are algebraic

over k, then

for some element q (which is algebraic over k). In other words, every ﬁnite extension

of a ﬁeld of characteristic 0 is simple.

Proof. See [Mill58].

B.11.5. Theorem. Let q be a transcendental element over a ﬁeld k and let K be a

ﬁeld such that k Ã K Õ k(q) with k π K. Then there exists an element s in K which is

transcendental over k and K = k(s).

Proof. See [Walk50].

qq q q

, ,...,

()

856 Appendix B Basic Algebra

Deﬁnition. Let f(X) Œ k[X] be a polynomial of positive degree. An extension ﬁeld K

of k is called a splitting ﬁeld of f(X) if

(1) f(X) is a product of linear factors in K{X}, that is,

with a Œ k, q

Œ K, and

(2) K = k(q

,...,q

B.11.6. Theorem. Every polynomial over a ﬁeld k of positive degree has a splitting

ﬁeld. Any two splitting ﬁelds of the polynomial are isomorphic over k.

Proof. See [Jaco64].

Deﬁnition. A polynomial f(X) Œ k[X] is said to be separable if it is a product of irre-

ducible polynomials each of which has only simple roots in a splitting ﬁeld of f(X).

B.11.7. Theorem. Every polynomial over a ﬁeld of characteristic 0 is separable.

Proof. See [Jaco64].

Deﬁnition. Let K be an extension of a ﬁeld k. An element of K is said to be separa-

ble over k if it is algebraic over k and its minimum polynomial is separable. K is a

separable extension of k if every element of K is separable over k.

Deﬁnition. Let K be an extension of a ﬁeld k. The elements a

, a

,..., a

in K

are said to be algebraically dependent over k if there exists a nonzero polynomial

f(X

,...,X

) in k[X

,...,X

] and f(a

,...,a

) = 0. Otherwise, the elements

are said to be algebraically independent. An inﬁnite set of elements of K is said to be

algebraically dependent over k if each of its ﬁnite subsets is algebraically dependent.

Otherwise it is said to be algebraically independent.

Alternatively, the elements a

, a

,..., a

are algebraically independent over k if

the map

is an isomorphism over k.

Deﬁnition. Let K be an extension of a ﬁeld k. A maximal set of elements in K that

are algebraically independent over k is called a transcendence basis for K over k. The

number of elements in a transcendence basis is called the transcendence degree of K

over k and is denoted by tr

(K).

The notion of transcendence degree is justiﬁed by the following fact:

B.11.8. Theorem. Any two transcendence bases of K over k have the same cardi-

nal number of elements.

kX X X ka a a

fX X X fa a a

12 12

, ,..., , ,...,

[]

()

fX aX X X

()

◊◊◊ -

()

qq q

B.11 Extension Fields 857

Proof. See [ZarS60].

Deﬁnition. A ﬁeld k is said to be algebraically closed if every nonconstant polyno-

mial f(X) in k[X] has a zero in k.

The complex numbers C are the standard example of an algebraically closed ﬁeld.

See Corollary E.5.2 for a proof.

Deﬁnition. Let K be an extension of a ﬁeld k. K is called an algebraic closure of k if

(1) K is algebraic over k and

(2) K is algebraically closed.

B.11.9. Theorem.

(1) Every ﬁeld k has an algebraic closure.

(2) Any two algebraic closures of k are isomorphic over k.

Proof. See [Jaco64].

B.11.10. Theorem. If k is an algebraically closed ﬁeld, then every polynomial in

k[X] factors into a product of linear factors.

Proof. See [Dean66].

B.11.11. Theorem. Every algebraically closed ﬁeld k is inﬁnite.

Proof. We use Theorem B.8.2. If k has characteristic 0, then the prime subﬁeld of k

is isomorphic to the reals. If k has characteristic p, p prime, then one needs to show

that the algebraic closure of Z

is inﬁnite.

B.11.12. Theorem. If k is an algebraically closed ﬁeld and f is a polynomial in

k[X

,...,X

] that vanishes on all but a ﬁnite subset of k

, then f is the zero

polynomial.

Proof. Because k is algebraically closed, it has an inﬁnite number of elements by

Theorem B.11.11. If n = 1, then this follows from Theorem B.8.9 since f can only have

a ﬁnite number of zeros. The general case is proved by induction on n.

Deﬁnition. If k is a ﬁeld, then k((X)) will denote the ﬁeld of quotients of k[[X]]. More

generally, k((X

,...,X

)) will denote the ﬁeld of quotients of k[[X

,...,X

]].

There are natural inclusions

The multiplicative inverse of a formal power series f in k[[X]] is 1/f in k((X)). The

following fact about 1/f is often used and is therefore worth stating explicitly, namely,

1/f is itself a power series in k[[X]] if it has a nonzero constant term.

kkX kXX kXX X

()()

Ã◊◊◊Ã

()()

112 12

, , ,..., .

858 Appendix B Basic Algebra