Section 9.1 Algebraic Structures
691
If i is the largest integer greater than 0 for which a i ∙ 0, the polynomial is of
degree i; if no such i exists, the polynomial is of zero degree. Terms with zero
coefficients are generally not written. Thus, px
4
−2∙3x
2
+ 5 is a polynomial of
degree 4, and the constant polynomial 6 is of zero degree. The set of all polynomials
in x over R is denoted by R3x 4.
We define binary operations of + and # in R3x 4 to be the familiar operations
of polynomial addition and multiplication. For polynomials f(x) and g(x) members
of R3x 4, the products f(x) # g(x) and g(x) # f(x) are equal because the coefficients are
real numbers, and we can use all the properties of real numbers under multiplication and addition (properties such as commutativity and associativity). Similarly,
for f (x), g(x), and h(x) members of R3x 4, ( f (x) # g(x)) # h(x) = f (x) # (g(x) # h(x)).
The constant polynomial 1 is an identity because 1 # f (x) = f (x) # 1 = f (x) for every f (x) [ R3x 4. Thus, 3R3x 4, # 4 is a commutative monoid. It fails to be a group
because only the nonzero constant polynomials have inverses. For example, there
is no polynomial g(x) such that g(x) # x = x # g(x) = 1, so the polynomial x has no
inverse. (Note that while x # (1∙x) = 1, 1∙x = x
−1
is not a polynomial.) However,
3R3x 4, + 4 is a commutative group.
PrACtiCe 8
a. For f (x), g(x), h(x) [ R3x 4, write the equations saying that R3x 4 under + is commutative
and associative.
b. What is an identity element in 3R3x 4, + 4?
c. What is an inverse of 7x
4
− 2x
3
+ 4 in 3R3x 4, + 4?
■
Polynomials play a special part in the history of group theory (the study of
groups) because much research in group theory was prompted by the very practical problem of solving polynomial equations of the form f (x) = 0, f (x) [ R3x 4.
The quadratic formula provides an algorithm for finding solutions for every f(x)
of degree 2, and the algorithm uses only the algebraic operations of addition, subtraction, multiplication, division, and taking roots. Other such algorithms exist
for polynomials of degrees 3 and 4. One of the highlights of abstract algebra is
the proof that no algorithm using only these operations exists for every f(x) of
degree 5. (Notice that this statement is much stronger than simply saying that no
algorithm has yet been found; it says to stop looking for one.)
The next example uses modular arithmetic. You may recall from Section 5.1
(Example 15 and the subsequent discussion) that each computer has some limit
on the size of the integers that it can store. Although we would like a computer
to be able to exhibit the behavior of 3Z, + 4, the best we can obtain is some finite
approximation. The approximation is achieved by performing addition modulo n.
The “answer” to the computation x + y for x, y [ Z is then either the actual value
x + y if this value falls within the limit that can be stored or a remainder value
obtained by doing modular arithmetic, which is equivalent to x + y under the
equivalence relation of congruence modulo n.
691
If i is the largest integer greater than 0 for which a i ∙ 0, the polynomial is of
degree i; if no such i exists, the polynomial is of zero degree. Terms with zero
coefficients are generally not written. Thus, px
4
−2∙3x
2
+ 5 is a polynomial of
degree 4, and the constant polynomial 6 is of zero degree. The set of all polynomials
in x over R is denoted by R3x 4.
We define binary operations of + and # in R3x 4 to be the familiar operations
of polynomial addition and multiplication. For polynomials f(x) and g(x) members
of R3x 4, the products f(x) # g(x) and g(x) # f(x) are equal because the coefficients are
real numbers, and we can use all the properties of real numbers under multiplication and addition (properties such as commutativity and associativity). Similarly,
for f (x), g(x), and h(x) members of R3x 4, ( f (x) # g(x)) # h(x) = f (x) # (g(x) # h(x)).
The constant polynomial 1 is an identity because 1 # f (x) = f (x) # 1 = f (x) for every f (x) [ R3x 4. Thus, 3R3x 4, # 4 is a commutative monoid. It fails to be a group
because only the nonzero constant polynomials have inverses. For example, there
is no polynomial g(x) such that g(x) # x = x # g(x) = 1, so the polynomial x has no
inverse. (Note that while x # (1∙x) = 1, 1∙x = x
−1
is not a polynomial.) However,
3R3x 4, + 4 is a commutative group.
PrACtiCe 8
a. For f (x), g(x), h(x) [ R3x 4, write the equations saying that R3x 4 under + is commutative
and associative.
b. What is an identity element in 3R3x 4, + 4?
c. What is an inverse of 7x
4
− 2x
3
+ 4 in 3R3x 4, + 4?
■
Polynomials play a special part in the history of group theory (the study of
groups) because much research in group theory was prompted by the very practical problem of solving polynomial equations of the form f (x) = 0, f (x) [ R3x 4.
The quadratic formula provides an algorithm for finding solutions for every f(x)
of degree 2, and the algorithm uses only the algebraic operations of addition, subtraction, multiplication, division, and taking roots. Other such algorithms exist
for polynomials of degrees 3 and 4. One of the highlights of abstract algebra is
the proof that no algorithm using only these operations exists for every f(x) of
degree 5. (Notice that this statement is much stronger than simply saying that no
algorithm has yet been found; it says to stop looking for one.)
The next example uses modular arithmetic. You may recall from Section 5.1
(Example 15 and the subsequent discussion) that each computer has some limit
on the size of the integers that it can store. Although we would like a computer
to be able to exhibit the behavior of 3Z, + 4, the best we can obtain is some finite
approximation. The approximation is achieved by performing addition modulo n.
The “answer” to the computation x + y for x, y [ Z is then either the actual value
x + y if this value falls within the limit that can be stored or a remainder value
obtained by doing modular arithmetic, which is equivalent to x + y under the
equivalence relation of congruence modulo n.