Why Read This Book? - Index of

304 Chapter 9 Rings
Furthermore, since p ≥ 2, it must be that b = d = 0, so that a =±1.
EXERCISE 9.4.4 Using i, j, k as in the quaternion group (Example 8.1.18),
construct the ring extension Z[i, j, k], defining equality, addition, and multiplication,
then showing that all ring properties R1–R10 are satisfied. Is it
commutative?
EXERCISE 9.4.5 Show that S ={a + 0i + cj + 0k : a, c ∈ R} is a subring of the
ring from Exercise 9.4.4.
EXERCISE 9.4.6 Finding the units in a ring amounts to solving the equation
xy = 1. In Z6, the only units are 1 and 5, so the equation xy ≡6 1 implies x, y ∈
{1, 5}. Use this fact and the technique in the proof of Theorem 9.4.3 to find all 16
units in Z6[ √ 2].
EXERCISE 9.4.7 Find, with verification, all units in Z[i].
EXERCISE 9.4.8 Prove that the following commutative rings with unity are
fields by showing that every nonzero element is a unit.
(a) Q[ √ 2]
(b) Q[i]
9.4.2 Polynomial Rings
The other type of extension we want to create might seem fundamentally different
from the previous ones, but the principle is the same. Its one notable difference
is that the new element we adjoin is, in a sense, more foreign to the original
ring than numbers like √ −5 are to the integers. The relationship of √ −5tothe
integers is characterized by the fact that ( √ −5) 2 =−5, which is an integer, or
if you prefer, ( √ −5) 2 �
+ 5 = 0. Similarly, if x = 1 + 3√ 2, then (x2 − 1) 3 − 2 = 0.
Thus, as with √ −5, there is some way to manipulate x using only the ring elements
and operations to produce zero. The term that describes this relationship of √ �
−5
and 1 + 3√ 2 to the integers is algebraic, and numbers that are not algebraic are
called transcendental. For example, π is transcendental over the integers because
there is no way to combine π and any finite set of integers using the ring operations
a finite number of times to produce zero. Strict definitions of these terms will come
in your later work in algebra. For now, we simply construct an example in which
the symbol we adjoin is transcendental over the integers because we define the
ring and the behavior of the symbol to make it so.
Let R be a commutative ring, and write R[t] to mean the set of all polynomials
in the variable t, where the coefficients are elements of R. That is,
R[t] ={ant n + an−1t n−1 +···+a1t + a0 : n ∈ W,ak ∈ R for all k,
(9.39)
and an �= 0ifn �= 0}