10. Preface
We consider mathematics as the tool kit of sciences from the practi-
cal side, and also a standalone discipline. Sciences acquire raw data by
observation from which conclusions are drawn by their own method-
ology, based on mathematics, and these conclusions, may as well be
called nature laws, are used for certain purposes, such as production of
something precious by procedures also governed by mathematics. Be-
sides applications it has also value per se, the courage to attack difficult
problems of human thinking. The quest for truth is performed by form-
ing precise concepts, statement of precise attributes of those concepts,
and establishing the statements by reasoning adherent strictly to logic.
We attempt to exhibit this to the student of mathematics, hopefully
with more success than failure.
Counting the rise of modern algebra from Leonard Euler, numer-
ous algebra books were written in numerous languages in the faith
that knowledge they acquired overcoming difficulties will be of use for
generations to come. However, sciences evolve in a rapid manner and
pages get dull. The content of pages to follow this preface has been
put down many times in several books in many ways, we still hope that
despite this, content will be of use for many students of algebra for at
least some decades to access knowledge not only in these pages but in
other books as well with more advanced or specialized topics with more
obscure essence.
To be brief, this book is aimed at mathematics or physics bachelor
majors. Although almost the whole matter may be used by IT bache-
lor majors while some parts may fit rather an IT master course. Two
CAS’s are used throughout the book, namely Maxima and GAP, ex-
ample scripts are included for reading together with exercises to write
scripts, forming an inherent part of the matter. For the future program-
mer more programming exercises are appropriate, and are welcome as
proposals, if there is interest at IT people, may be included in a later
v
11. edition. Anyway, we did not have an IT student in mind while algo-
rithmic approach is applied at several points in the matter. For other
science disciplines a restricted use may occur. If not a future specialist,
with time some parts of the material may get obscured. Training of
the reader’s talent should remain as a valuable asset after algebra or
introductory mathematics course we hope.
Mathematics is one of the most profitable subjects to pursue from
many aspects. Holding a degree in mathematics or physics is a great
asset in job seeking indisputably. Moreover, she is taught to science,
economics and IT students, who, possibly unconsciously, utilize her at
least indirectly in solving problems of their own professions at every
day of their career. Understanding her concepts and statements one
communicates with the greatest minds of the millennia past and to
come as her results were accumulated since the dawn of mankind and
will be utilized until the fall of human thinking. Mathematics was and
will remain the main impetus behind advancement.
This is the first part of a set planned two volume, dealing with
elementary topics. We wish to provide continuity in learning for the
freshman. After a thorough secondary mathematics study it is hard
to grasp why further. Usefulness of material is to be emphasized as
hardships are enormous due to great number of involved concepts and
elaborate procedures. This is incompatible with the labelling at first
glance. A sound elementary mathematics routine is expected from
secondary school, here the label ’elementary’ refers to secondary. In
the label ’elementary’ of the first part refers to bachelor level which
is the base of master and doctorate levels, in our viewpoint subject
matter of the first part acts as an intermediary to more advanced levels.
This division is somewhat mannered considering unity of the discipline,
which is rarely untangled as vastness of discipline prevails.
This vastness comes forward already in the first part as at certain
points it is almost impossible to keep linearity of subject matter as
one may not eschew reference to a later section or advancement of a
definition at an early stage not fitting to the present context. Therefore
repetitions may befall but only occasionally.
An emphasis has been put on exercises. In this case there is allure-
ment to be lazy with core matter to give forth non-rigorous definitions
and sketchy proofs. How great are the odds in favour of being practi-
cal, we endure throughout the text by giving forth exact content un-
dertaking complexity if unavoidable. These exercises come from many
sources, mainly from personal course matters ranging from the triv-
vi
12. ial to the almost inextricable without a hint. We did not keep order
of difficulty but the inextricable type is rare, almost every exercise is
within reach of the student. Answers are absent because of the limi-
tation in extent, it is planned that readers provide solutions through
the website of the book encouraging communication between readers,
providing means to even submitting new exercise proposals to include
in later edition.
What is included is standard material, perhaps more detailed at cer-
tain points than usual. The book encompasses courses of more terms
with various labels at institutions. As scope may vary additional ma-
terial is possibly needed, although an attempt has been made to be as
self-contained as possible.
Maybe it is superfluous to give instructions for use here as the in-
structor or the learner should decide the pace, what to skip or perhaps
what supplement to include live at a lecture or seminar. The order of
the material is bound as notions and assertions are used at later stages
frequently, interconnections form such a thick web that it is almost im-
possible to change order of chapters without harm to consistency. The
attribute ’almost’ appears since with proper caution some re-ordering
may be implemented.
János Kurdics
vii
16. Chapter 1
Introduction into algebra
1.1 Basic notions
1.1.1 Sets
The most basic conceptNotions: set,
universe,
empty set,
singleton,
element
of mathematics is that of the set, which
intuitively means an aggregate of things. Below we are going to discuss
the so called naïve set theory, a rigorous substantiation belongs to
the scope of mathematical logic. One begins with the set universe,
including all things examined. If A is a set then it has an element
a (notation a ∈ A) unless it is the empty set (notation ∅) for which
there is not any object a with a ∈ A. A set with a single element is
called a singleton. Of a set it is assumed that for every object a one
can determine whether a ∈ A or a /∈ A, the latter meaning that a is
not an element of the set. It also assumed that among the elements of
the universe there are not any sets.
In mathematics researchNotions: set
of natural
numbers,
natural
number,
Peano’s
axioms
usually the universe or a part of the uni-
verse is the set of natural numbers, existence of which is supposed
(notation N). Elements of this set are the natural numbers, and the
so called Peano’s axioms hold:
1. there exists an element called zero of the set N of natural numbers
(notation 0);
2. for any natural number a there exists a natural number a called
the consecutive of a, satisfying the following properties:
3. distinct natural numbers have distinct consecutives;
3
17. CHAPTER 1. INTRODUCTION INTO ALGEBRA
4. there does not exist a natural number with consecutive zero;
5. (axiom of mathematical induction) if A is a set consisting of natu-
ral numbers, the set A contains zero and with each of its elements
the set A contains its consecutive then the sets A and N shall
coincide.
Usual denominations and notations: 0’=1 one, 1’=2 two, 2’=3 three
and so on.
The key to the method of recursive definition Notions:
recursive
definition,
inductive
proof
is the last axiom
of Peano. Notions or elements involving a natural number parameter
are determined for any natural number by defining the one belonging
to zero, and then, assuming we have already known the one belonging
to some natural number n, by defining the notion or element belonging
to natural number n . The key to the method of inductive proof is,
as well, the last axiom of Peano. An assertion depending on a natural
number as a parameter is proved for any natural number by showing
the one belonging to zero, and then, assuming the truth of the one
belonging to some natural number n, by showing that the assertion
belonging to natural number n follows.
The power set of a set B Notions:
subset, (strict)
inclusion,
power set, ,
equality of
sets,
determination
of sets
(notation P(B)) is also a set, consisting
of all the subsets A of the set B, that is, all such sets A, every element
of which is an element of the set B (notation A ⊆ B„ we also say that
the set B includes the set A). Two sets A and B are equal if
mutually includes the other (notation A = B). The set A is strictly
contained in or a proper subset of the set B if B includes A
but they are not equal (notation A ⊂ B). One can determine the
set A by enumeration such as A = {a1, a2, a3, a4}, or by specification
by virtue of a property P among the elements of the set B such as
A = {a ∈ B|P(a)}, meaning that from the elements of the set B
exactly those elements form the set A that satisfy the property P.
Consider now set operations Notions: set
union,
intersection,
difference,
disjoint sets,
complement of
a set,
symmetric
difference,
union and
intersection of
a family of
sets
(the exact concept of algebraic op-
erations and their properties will be considered below). Let A and B
sets. Their union, A ∪ B is the set containg exactly those elements
which are elements of the set A or elements of the set B. Their inter-
section, A ∩ B is the set containg exactly those elements which are
elements of the set A and elements of the set B. Their difference,
A B is the set containg exactly those elements which are elements of
the set A but not elements of the set B. If the intersection of two sets
is the empty set then we say that the sets are disjoint. If the set A is
included in the set B then the complement of A with reference to the
4
18. Chapter 2
Introduction of number
concept
2.1 Natural and whole numbers
2.1.1 Natural numbers
Notions:
addition,
multiplication
Recall that in 1.1.1 Peano’s axioms were used to define the no-
tions of natural numbers and the set of natural numbers. Operations
addition and multiplication of natural numbers are defined by
recursion for an arbitrary natural number a ∈ N in the following man-
ner: let a + 0 = a, a · 0 = 0, for an arbitrary natural number b ∈ N let
a + b = (a + b) and ab = ab + a, where a is called the consecutive of
a. Method of proving the next theorem is induction.
Operations
on natural
numbers (i) (N, +) is a commutative monoid and every element can be can-
celled;
(ii) (N, ·) is a commutative monoid and every nonzero element can be
cancelled;
(iii) multiplication is distributive with respect to addition.
Proof Let a, b and c natural numbers.
(i) Associativity. Clearly, (a+b)+0 = a+b = a+(b+0), and assume
(a + b) + c = a + (b + c). Then by applying definition and inductive
assumption, (a + b) + c = ((a + b) + c) = a + (b + c) = a + (b + c ).
39
19. CHAPTER 2. NUMBER CONCEPT
Neutral element. According to definition, a + 0 = a holds. Clearly,
0 + 0 = 0, and suppose 0 + a = a. Then 0 + a = (0 + a) = a .
Commutativity. As 0 is the neutral element, a + 0 = 0 + a, and
assume a + b = b + a. Then a + b = (a + b) = (b + a) = b + a ,
and it suffices to show b + a = b + a, by induction again. We have
b + 0 = (b + 0) = b = b + 0, and assume b + a = b + a. Then
b + a = (b + a ) = (b + a) = b + a .
Cancellation. By commutativity it suffices to show right cancella-
tion. As 0 is the neutral element, a + 0 = b + 0 implies a = b. Suppose
a + c = b + c implies a = b. Let a + c = b + c ; then (a + c) = (b + c) ,
and because consecutivity is injective, we have a + c = b + c, which
follows a = b by induction.
To establish statement (ii) one applies similar inductive arguments.
Distributivity is easy to prove: a(b + 0) = ab = ab + 0 = ab + a0 holds,
and suppose a(b + c) = ab + ac. It follows a(b + c ) = a(b + c) =
a(b + c) + a = (ab + ac) + a = ab + (ac + a) = ab + ac . Commutativity
of multiplication follows (a + b)c = ac + bc. Q.E.D.
Notion:
ordering of
natural
numbers
We say that the natural number a is less than or equal to the
natural number b if for some natural number c, b = a+c holds, notation:
a ≤ b, and of sharp inequality, a < b. This is a linear ordering on N,
and any nonempty subset of N contains an infimum, which is 0 for the
whole N.
Exercises
1. Prove that multiplication of natural numbers is associative.
2. Prove that 0 is a neutral element with respect to multiplication
of natural numbers.
3. Prove that multiplication of natural numbers is commutative.
4. Prove that multiplication of nonzero natural numbers is cancella-
tive.
5. Prove that (usual) ordering of natural numbers is a linear order
with minimal element 0.
6. Prove that every nonempty subset of natural numbers contains
an infimum.
7. Show 1 + 3 + 5 + · · · + (2n − 1) = n2
(n ∈ N+
).
40
20. Chapter 3
Elementary linear algebra
3.1 Free vectors
3.1.1 Space free vectors and their operations
Notions:
direction of a
ray, directed
line segment ,
initial point,
endpoint,
congruent
directed line
segments
Two rays of the (classical sense) euclidean space are said to have
the same direction if either coincide or are parallel and are contained
in the same half-plane determined by the straight line connecting the
initial points. Clearly, this is an equivalence relation on the set of rays
of the space. Ordered pairs of points of the space are called directed
line segments, the first term of the pair is called the initial point, the
second term the endpoint. The directed line segments (A, B), (C, D)
are called congruent if either A = B and C = D, or the lengths
AB = CD, and the rays
−→
AB and
−−→
CD have the same direction.
Congruence
of directed
line
segments
We see
readily the next assertion.
Congruence of directed line segments of the space is an equivalence
relation.
Notions:
free vector in
space, origon,
magnitude of
the free
vector, zero
vector,
direction of a
vector,
position
vector
An equivalence class of congruent directed line segments in space is
called a (space) free vector, notation AB for the class represented by
the directed line segment (A, B), or frequently a for the the class rep-
resented by the directed line segment (O, B), where O is a fixed point
in space often referred to as the origon. Let the set of all space free
vectors be denoted by V. The free vector represented by the directed
line segment (A, A) for some (and any) point A in space is called the
zero vector denoted by o. Length AB of the line segment AB is called
the magnitude of the free vector AB, notation |AB|. Direction of
87
21. CHAPTER 3. ELEMENTARY LINEAR ALGEBRA
the ray
−→
AB is called the direction of the nonzero free vector AB.
To the zero vector we do not assign a direction, or sometimes we say
that its direction is arbitrary. Magnitude and direction is clearly inde-
pendent of choice of representative. We hence may also say that two
free vectors equal if their magnitudes and directions coincide. We see
that a free vector may be represented by a directed line segment with
arbitrary initial point, hence free, and after fixing the initial point, the
endpoint is unique. In geometry it is often useful to fix a space point
origon O and then identify a point A of space with the free vector
a = OA often called a position vector.
Sum of two space free vectors a and b is determined as follows.
Represent the free vectors such that the endpoint a coincide with the
initial point of b i.e. a = AB and b = BC. Then let a + b = AC.
Let λ ∈ R be nonnegative, AB a free vector. Scalar multiple of
the free vector AB by λ ≥ 0 is the free vector λAB = AC such that
C lies on the ray
−→
AB and has magnitude |AC| = λ|AB|. Let λ ∈ R be
negative, AB a free vector. Scalar multiple of the free vector AB
by λ < 0 is the free vector λAB = AC such that C lies on the opposite
ray of
−→
AB and has magnitude |AC| = |λ| · |AB|.
Free
vector
operation
properties
(i) (V, +) is an Abelian group;
(ii) for any λ ∈ R and a, b ∈ V, λ(a + b) = λa + λb;
(iii) for any λ, μ ∈ R and a ∈ V, (λ + μ)a = λa + μa;
(iv) for any λ, μ ∈ R and a ∈ V, (λμ)a = λ(μa);
(v) for any a ∈ V, 1a = a és 0a = o.
Proof(i) Addition is well-defined. Let a = AB = DE and b = BC = EF.
We have to show AC = DF. If A = B or B = C or C = A then it is
obvious. In the contrary case the lengths of the directed line segments
(A, B) and (D, E) coincide and the rays
−→
AB and
−−→
DE have the same
direction, and also the lengths of the directed line segments (B, C) and
(E, F) coincide and the rays
−−→
BC and
−→
EF have the same direction.
Consequently, the triangles ABC and DEF are congruent, their
88
22. 3.1. FREE VECTORS
(ii) b = p + q and p is parallel to c
(iii) c = p + q and p is parallel to d
(iv) d = p + q and p is parallel to e
(v) e = p + q and p is parallel to f
(vi) f = p + q and p is parallel to a
22. Check validity of the cosine theorem for the angle at vertex with
position vector
(i) a in the triangle with other vertices with position vectors b
and c
(ii) a in the triangle with other vertices with position vectors b
and d
(iii) a in the triangle with other vertices with position vectors d
and e
(iv) b in the triangle with other vertices with position vectors c
and d
(v) b in the triangle with other vertices with position vectors c
and e
(vi) b in the triangle with other vertices with position vectors c
and f.
23. Determine the area of the parallelogram spanned by the vectors
(i) a and b; (ii) b and e; (iii) c and d; (iv) a and f; (v) c and e;
(vi) b and f.
24. Determine the volume of the tetrahedron spanned by the vectors
(i) a, b and d; (ii) b, c and e; (iii) c, d and f; (iv) a, e and f; (v)
c, d and e; (vi) b, e and f.
25. Check your computations by Maxima. For instance, the compu-
tations (vii) and (ix):
load(vect);
d:[-2,3,1];
e:[1,2,-2];
f:[1,-1,1];
d . ((-2)*e+3*f);
a:[1,-1,2];
express(express(f~e)~a);
26. Furnish a Maxima function for mixed product.
99
23. CHAPTER 3. ELEMENTARY LINEAR ALGEBRA
3.1.2 Coordinate geometry
Notions:
normal vector
of a plane,
direction
vector of a
straight line
For coordinate geometry considerations fix an origon O and identify
space points A with position vectors OA = a. In this respect we may
consider vectorial equations of space elements. Normal vector of a
plane is a nonzero free vector perpendicular to the plane, Direction
vector of a straight line is a non-zero vector parallel with the straight
line.
Equations
of space
elements
(i) Let the plane be given by its point A with position vector a = OA
and by its normal vector n, let an arbitrary point P of the plane
have position vector p = OP. Then the normal vectorial equation
of the plane is n(p − a) = 0.
(ii) Let the plane be given by its point A with position vector a = OA
and by two linearly independent vectors u and v parallel to the
plane, and let an arbitrary point P of the plane have position
vector p = OP. Then the parametric equation of the plane is
p = a + ru + sv (r, s ∈ R).
(iii) Let the straight line be given as the intersection line of two inter-
secting planes. Let an arbitrary point P of the plane have position
vector p = OP, then the system of equations of the straight line
is
n(p − a) = 0
n (p − a ) = 0,
where the equations are normal equations of the two intersecting
planes.
(iv) Let the straight line be given by its point A with position vector
a = OA and by its direction vector u. Let an arbitrary point
P of the plane have position vector p = OP, then the direction
vectorial equation of the straight line is p = a + ru, where r ∈ R.
Proof(i) The inner product n(p − a) vanishes if and only if p − a is
perpendicular to n, which holds if and only if p − a is parallel with the
plane i.e. p is the position vector for some point lying on the plane.
(ii) The vector ru + sv is parallel to the plane hence a + ru + sv is the
100
24. 3.1. FREE VECTORS
O
A
P
x
n
O
x
A
u
v
Px
position vector for some point lying on the plane. If p is the position
vector for some point lying on the plane then p − a is parallel to the
plane hence is a linear combination of the linearly independent vectors
u and v.
(iii) Both normal equations are satisfied by position vectors leading
to points of the intersection line and only by those.
A
P
x
x
u
(iv) As a is the position vector of a point of the line and ru is
parallel with the line, a + tu is the position vector of a point lying on
the line. If p is the position vector of a point lying on the line then
101
25. CHAPTER 3. ELEMENTARY LINEAR ALGEBRA
p−a is parallel to the line and hence is a scalar multiple of the direction
vector u. Q.E.D.
Fix an orthonormal basis {ei}i=1,2,3, and identify points of the space
with triples of coordinates of their position vectors.
Notions:
scalar
equation
(parametric
system of
equations) of
the plane,
scalar
(parametric,
canonical)
system of
equations of
the straight
line
Consider normal vector equation n(p − a) = 0 of the plane, and let
n = n1e1 + n2e2 + n3e3, p = xe1 + ye2 + ze3, a = a1e1 + a2e2 + a3e3.
Then
0 = n(p−a) = (n1e1 +n2e2 +n3e3)((x−a1)e1 +(y−a2)e2 +(z−a3)e3) =
n1x + n2y + n3z − (n1a1 + n2a2 + n3a3)
i.e. scalar equation of the plane is n1x+n2y +n3z −(n1a1 +n2a2 +
n3a3) = 0. Consider parametric equation p = a + ru + sv of the plane,
and let u = u1e1 +u2e2 +u3e3, v = v1e1 +v2e2 +v3e3, p = xe1 +ye2 +ze3,
a = a1e1 + a2e2 + a3e3. Then
xe1+ye2+ze3 = (a1+ru1+sv1)e1+(a2+ru2+sv2)e2+(a3+ru3+sv3)e3
which gives, by uniqueness of coordinates, parametric system of
equations of the plane:
x = a1 + ru1 + sv1
y = a2 + ru2 + sv2
z = a3 + ru3 + sv3.
Scalar system of equations of the straight line has form
n1x + n2y + n3z − (n1a1 + n2a2 + n3a3) = 0
n1x + n2y + n3z − (n1a1 + n2a2 + n3a3) = 0.
Consider direction vector equation p = a + ru of the straight line, and
let u = u1e1 + u2e2 + u3e3, p = xe1 + ye2 + ze3, a = a1e1 + a2e2 + a3e3.
Then
xe1 + ye2 + ze3 = (a1 + ru1)e1 + (a2 + ru2)e2 + (a3 + ru3)e3
which gives, by uniqueness of coordinates, parametric system of
equations of the straight line:
x = a1 + ru1
y = a2 + ru2
z = a3 + ru3.
102
26. Chapter 4
Elementary polynomial
theory
4.1 Ring of polynomials
4.1.1 The integral domain A[x]
Notions:
indeterminate,
coefficient,
univariate
polynomial,
degree, monic
polynomial,
equality of
polynomials,
value of f(x)
at c
Let A be a commutative ring with unity, x /∈ A a symbol called
indeterminate. The formal sum f(x) = anxn
+an−1xn−1
+· · · a1x+a0
with the coefficients ai ∈ A, an = 0 or every coefficient zero, n ∈ N is
called a univariate polynomial over A, n, if not all coefficients are
zero, is called the degree of the polynomial denoted by f◦
. If an = 1
then f(x) is called a monic polynomial. Two polynomials are equal
if their respective coefficients coincide. If c ∈ A then, substituting c
in f(x) in place of x, considering the formal operations in f(c) ring as
operations in A, the value f(c) ∈ A is called value of f(x) at c.
Addition and multiplication of polynomials are performed by ap-
plying all the usual properties. It is not hard to see that the structure
obtained is a commutative ring with unity. In formal way we obtain
the following.
Structure
of
polynomials
Let A be a commutative ring with unity. Denote by A[x] the set of
all sequences of elements of A with only finitely many terms nonzero.
Addition and multiplication for any a, b ∈ A[x] are defined as follows:
(a + b)n = an + bn, (ab)n =
n
i=0
aibn−i.
147
27. CHAPTER 4. ELEMENTARY POLYNOMIAL THEORY
Then (A[x], +, ·) is a commutative ring with unity, moreover, if A is
an integral domain then A[x] is also an integral domain.
ProofFor, let a, b, c ∈ A[x]. Obviously, addition and multiplication are
algebraic operations.
Additive structure is an Abelian group. This is immediate from the
respective properties of the additive structure of A.
Multiplicative structure is associative. On one hand,
((ab)c)n =
n
i=0
(ab)icn−i =
n
i=0
i
j=0
ajbi−jcn−i,
on the other hand
(a(bc))n =
n
i=0
ai(bc)n−i =
n
i=0
n−i
j=0
aibjcn−i−j.
Both expressions obtained coincide with the sum i+j≤n aibjcn−i−j.
Commutativity of multiplication is obvious. The sequence with ze-
roth term 1 and all the other 0 is clearly a unity.
Distributivity. By commutativity it suffices to prove one of the two
properties. We see
y((a + b)c)n =
n
i=0
(a + b)icn−i =
n
i=0
(ai + bi)cn−i =
n
i=0
(aicn−i + bicn−i) =
n
i=0
aicn−i +
n
i=0
bicn−i = (ac)n + (bc)n.
Zero divisor freeness. Let a and b be two not constant zero sequence
with nonzero terms of maximal index an and bm. Then (ab)n+m =
anbm = 0 by zero divisor freeness of A. The proof is complete. Q.E.D.
Notions:
convolution
product,
polynomial
ring
Such a product of sequences is called convolution product . Iden-
tify with a ∈ A the sequence with zeroth term a all the other 0, identify
with x the sequence with first term 1 all the other 0; and so on, with xn
the sequence with nth term 1 all the other 0 and so on. In this manner
the formal sum f(x) = anxn
+ an−1xn−1
+ · · · + a1x + a0 becomes an
expression applying ring operations of A[x], and every sequence in the
set A[x] may be expressed in this form, from now on polynomials are
considered as of form f(x) = anxn
+ an−1xn−1
+ · · · + a1x + a0, the
structure A[x] is called the univariate polynomial ring over the
ring A. We shall assume below that in case of the ring of polynomials
A[x], A is an integral domain.
148
28. CHAPTER 4. ELEMENTARY POLYNOMIAL THEORY
g(x) = bn(x − y1)(x − y2) · · · (x − yn).
The resultant vanishes if and only if there is a common root of the
polynomials in the field L, moreover, by Viéte’s formulae and the fun-
damental theorem on symmetric functions the resultant is an
mbm
n times
a symmetric function of the roots. Consider xi and yj as indetermi-
nates. If xi = yj a common root then the resultant vanishes identically
hence the resultant is divisible by xi − yj. As the indices i and j are
arbitrary it follows that the resultant is divisible by
S = an
mbm
n
m
i=1
n
j=1
(xi − yj).
As f(x) = am
m
i=1(x − xi), S = (−1)mn
bm
n
n
j=1 f(yj). Consequently
monomials in the polynomial S are of degree n in the ai. Similarly, as
g(x) = b0
n
j=1(x − yj),
S = an
m
m
i=1
g(xi).(∗)
Consequently monomials in the polynomial S are of degree m in the
bj. Since in the summands of the resultant R the degrees are the same
and S divides R, necessarily the polynomials R and S are associates
in the ring K[x, y] hence differ in a nonzero scalar factor. Consider
these as polynomials of b0. By the formula S = an
m
m
i=1 g(xi) and the
determinant for of R we see that both S and R the highest degree m
term in b0 has coefficient +an
m. Therefore R = S. Summing up:
Resultant
in terms of
roots
Let K be a field, f(x), g(x) ∈ K[x] of degrees m and n (m, n ≥
1) with leading coefficients am and bn, respectively. Their resultant
is R = an
mbm
n
m
i=1
n
j=1(xi − yj) where xi and yj are the roots of the
polynomials.
We establish the connection between resultant and discriminant.
Resultant
and
discriminant
Let the polynomial
f(x) = amxm
+ am−1xm−1
+ · · · + a1x + a0
of degree m (m ≥ 2) over the field K have discriminant D, and
let f(x) and its derived function f (x) have resultant R. Then R =
(−1)
m(m−1)
2 amD.
ProofFor, let f(x) have roots x1, x2, . . . , xm in some field L having K
as a subfield. . (This condition may always be satisfied). By (*)
190
29. 4.4. MULTIVARIATE POLYNOMIALS
R(f, f ) = am−1
m
m
i=1 f (xi), and applying the rule of product polyno-
mial derivation
f (x) = am
m
i=1
(x − x1) · · · (x − xi−1)(x − xi+1) · · · (x − xm),
which follows
f (xi) = am(xi − x1) · · · (xi − xi−1)(xi − xi+1) · · · (xi − xm).
We conclude
R(f, f ) = a2m−1
m
i=j
(xi − xj) = (−1)
m(m−1)
2 amD.
Q.E.D.
Notice that the statement shows that the discriminant is a polyno-
mial of the coefficients ai as in the determinant form of the resultant
the coefficient am may be factored out.
Exercises
1. Determine the discriminant.
(i) x3
− 3 x2
+ 2 x − 1
(ii) x3
+ 3 x2
+ 2 x + 1
(iii) x3
+ 2 x2
+ 1
(iv) x3
− 2 x2
+ 2
(v) x3
− 2 x + 2
(vi) x4
+ 2 x3
− 2 x − 1
2. Solve the systems of equations by the method of resultants.
(i)
x2
− xy − 2y2
− 1 = 0
x2
+ y2
− 1 = 0
}
(ii)
(y − 1)x2
+ (y + 2)x + (y + 5) = 0
(y + 1)x2
+ (y − 1)x + (y − 9) = 0
}
(iii)
(y − 2)x2
+ (y − 3)x + (y − 2) = 0
yx2
+ (y + 3)x + (y + 2) = 0
}
(iv)
x2
+ (y + 1)x + (y − 3) = 0
x2
+ yx + (y − 5) = 0
}
(v)
x2
+ (y − 1)x + (y − 2) = 0
(y − 2)x2
+ yx + (y − 1) = 0
}
191
30. CHAPTER 4. ELEMENTARY POLYNOMIAL THEORY
(vi)
xy2
+ xz = 0
x + y + z = 0
}
(vii)
4x2
+ (−7y + 13)x + (y2
− 2y − 3) = 0
9x2
+ (−14y + 28)x + (y2
− 4y − 5) = 0
}
(viii)
x2
− 3x + (y2
− y) = 0
−x2
+ (−6y + 7)x + (y2
+ 11y − 12) = 0
}
3. Check your calculations by Maxima using the built-in functions
resultant, diff and solve.
192
31. Bibliography
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Tankonyvkiado, Budapest, 1980. in Hungarian, textbook.
[2] G. Chrystal. Algebra – an elementary textbook I. Adam and
Charles Black, London, Edinburgh, 1893.
[3] Sominskii, I.S., Faddeev, D.K. Problems in Higher Algebra. Books
in Mathematics. W.H. Freeman, New York, 1965.
[4] L. Fuchs. Algebra. Tankonyvkiado, Budapest, 1981. in Hungarian,
textbook.
[5] The GAP Group. GAP – Groups, Algorithms, and Programming,
Version 4.7.5, 2014.
[6] I. Reiman, F. Gyapjas. Exercises from elementary mathematics
vol I. Tankonyvkiado, Budapest, 1991. in Hungarian, textbook.
[7] P.R. Halmos. Lectures on Boolean Algebra. Van Nostrand, Prince-
ton, 1963.
[8] J. Kurdics. Algebra basics. Bessenyei Konyvkiado, Nyiregyhaza,
2006. in Hungarian, textbook.
[9] J. Kurdics. Discrete mathematics. Bessenyei Konyvkiado, Nyire-
gyhaza, 2006. in Hungarian, textbook.
[10] J. Kurdics. Algebra I. Bessenyei Konyvkiado, Nyiregyhaza, 2007.
in Hungarian, textbook.
[11] J. Kurdics. Algebra II. Bessenyei Konyvkiado, Nyiregyhaza, 2007.
in Hungarian, textbook.
193
32. [12] Maxima. Maxima, a computer algebra system. version 5.27.0,
2012.
[13] T. Frey., P. Pachné. Vector and tensor calculus. Muszaki Konyvki-
ado, Budapest, 1970. in Hungarian.
[14] S. Róka. 2000 exercises around elementary mathematics. Typotex,
Budapest, 2000. in Hungarian.
[15] K.H. Rosen et al Handbook of discrete and combinatorial mathe-
matics. CRC Press, Boca Raton, 1999.
[16] V. Scharnitzky. Matrix calculus. Muszaki Konyvkiado, Budapest,
1973. in Hungarian.
[17] G. Szász. Lattice theory. High school booklets. Tankonyvkiado,
Budapest, 1978. in Hungarian.
[18] J. Szendrei. Algebra and number theory. Tankonyvkiado, Bu-
dapest, 1978. in Hungarian, textbook.
[19] B.L. van der Waerden. Algebra I, II. Springer Verlag, New York,
Berlin, Heidelberg, 2003.
194
39. PROPOSITION INDEX
Proposition index
A
Algebraic property of matrix prod-
uct . . . . . . . . . . . . . . . 114
B
Bézout’s theorem . . . . . . . . . 158
Basic properties of divisibility
. . . . . 69, 149
Basic properties of linear map-
pings . . . . . . . . . . . . . 138
Basic properties of matrix rep-
resentation . . . . . . . 143
Basis for free vectors . . . . . . 91
Binomial theorem . . . . . . . . . 57
C
Cardano’s formulae . . . . . . . 170
Cardinality of primes . . . . . . 77
Characterization of multiplicity
k . . . . . . . . . . . . . . . . . 176
Characterization of polynomials
not relatively primes . .
. . . . . . . 188
Characterizations of basis 109
Cofactor expansion . . . . . . . 123
Common root and g.c.d. . . 177
Composition Bracketing . . . . 9
Congruence of directed line seg-
ments . . . . . . . . . . . . . . 87
Conversion from impure to com-
mon decimal . . . . . . . 83
Conversion from pure to com-
mon decimal . . . . . . . 82
Coordinates and transformation
under basis change 145
Corollary to Fundamental theo-
rem of algebra . . . . 167
Corollary to root factor decom-
position . . . . . . . . . . 160
Cramer’s rule . . . . . . . . . . . . 134
Crossed product properties 94
D
D.n.f. and c.n.f . . . . . . . . . . . . 31
Decimal fraction forms of ratio-
nals . . . . . . . . . . . . . . . 83
Determinant properties . . . 120
Direct complement . . . . . . . 110
Disjoint cycle decomposition the-
orem . . . . . . . . . . . . . . 17
Distance of space elements 103
E
Eisenstein’s theorem . . . . . . 163
Equations of space elements . .
. . . . . . . 100
Equivalence and Partitioning 8
Euclidean division . . . . 70, 150
Existence of basis . . . . . . . . 109
Existence of l.c.m. . . . . 73, 152
Extended Euclidean algorithm
. . . . . 71, 152
F
Finest direct decomposition . . .
. . . . . . . 110
Finite Boolean algebras . . . . 29
First theorem on partial fractions
. . . . . . . 156
Free vector operation properties
. . . . . . . 88
Functional completeness . . . 32
Fundamental theorem of algebra
. . . . . . . 167
201
40. Fundamental theorem of num-
ber theory . . . . . . . . . 76
Fundamental theorem of poly-
nomial theory . . . . 154
Fundamental theorem on sym-
metric polynomials 185
G
G.c.d. and resultant . . . . . . 189
G.c.d. by Euclidean algorithm
. . . . . 70, 151
H
Horner’s scheme . . . . . . . . . . 159
I
Identities of exponentiation 55
Implications of Associativity 15
Implications of Distributivity .
. . . . . . . 25
Inner product properties . . . 93
Invertibility and determinant .
. . . . . . . 134
Invertibility of Functions . . . . 9
K
Known roots of a reciprocal 178
Kronecker–Capelli’s Theorem .
. . . . . . . 133
L
Laws of De Morgan . . . . . . . . 27
Lemma on dependence . . . 108
M
Matrix of product transforma-
tion . . . . . . . . . . . . . . 144
Matrix product and matrix ad-
dition . . . . . . . . . . . . 115
Matrix rank theorem . . . . . 133
Maximal Boolean ideals . . . 28
Mixed product properties . . 96
N
Newton’s formulae I . . . . . . 182
Newton’s formulae II . . . . . 182
Nullity plus rank . . . . . . . . . 139
Number systems . . . . . . . . . . . 79
O
Operations in polar form . . 66
Operations on natural numbers
. . . . . . . 39
P
Polynomial decomposing lemma
. . . . . . . 156
Polynomial theorem . . . . . . . 57
Posets and lattices . . . . . . . . . 26
Prime and irreducible polynomi-
als over a field . . . . 154
Prime and noncompound num-
bers . . . . . . . . . . . . . . . 75
Prime power in factorial . . . 80
Product theorem of determinants
. . . . . . . 124
Properties of absolute value 64
Properties of complex conjuga-
tion . . . . . . . . . . . . . . . . 63
Properties of g.c.d. . . . . 72, 152
Properties of l.c.m. . . . 73, 152
R
Rank and elementary transfor-
mations . . . . . . . . . . 132
Real cubic irreducible case 172
Real cubic positive case . . 171
Real cubic zero case . . . . . . 171
Real decimal fractions . . . . . 84
Reciprocal after eliminating known
root factors . . . . . . . 179
Reciprocal equations and coeffi-
cients . . . . . . . . . . . . . 178
Resultant and discriminant 190
202
41. PROPOSITION INDEX
Resultant in terms of roots 190
Rolle’s theorem . . . . . . . . . . . 161
Root factor decomposition 160
Roots of a polynomial and its
derived polynomial 175
S
Second theorem on partial frac-
tions . . . . . . . . . . . . . 157
Set fields . . . . . . . . . . . . . . . . . . 28
Skew expansion . . . . . . . . . . 133
Space of linear mappings . 140
Stone’s Theorem . . . . . . . . . . 30
Structure of m×n matrices 114
Structure of a multivariate poly-
nomial ring . . . . . . . 181
Structure of complex numbers
. . . . . . . 62
Structure of integers . . . . . . . 42
Structure of linear transforma-
tions . . . . . . . . . . . . . 140
Structure of polynomials . 147
Structure of quadratic matrices
. . . . . . . 116
Structure of rationals . . . . . . 46
Structure of reals . . . . . . . . . . 51
Structure of symmetric polyno-
mials . . . . . . . . . . . . . 184
T
Theorem on field of quotients .
. . . . . . . 48
Theorem on multiple roots 177
U
Units of A[x] . . . . . . . . . . . . . 149
V
Viéte’s formulae . . . . . . . . . . 162
203
42.
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