Fields

In what follows we shall have occasion to use various classes of numbers (such as the class of all real numbers or the class of all complex numbers). Because we should not, at this early stage, commit ourselves to any specific class, we shall adopt the dodge of referring to numbers as scalars . The reader will not lose anything essential if he consistently interprets scalars as real numbers or as complex numbers; in the examples that we shall study both classes will occur. To be specific (and also in order to operate at the proper level of generality) we proceed to list all the general facts about scalars that we shall need to assume.

(A) To every pair, and , of scalars there corresponds a scalar , called the sum of and , in such a way that

addition is commutative, ,
addition is associative, ,
there exists a unique scalar (called zero ) such that for every scalar , and
to every scalar there corresponds a unique scalar such that .

(B) To every pair, and , of scalars there corresponds a scalar , called the product of and , in such a way that

multiplication is commutative, ,
multiplication is associative, ,
there exists a unique non-zero scalar (called one ) such that for every scalar , and
to every non-zero scalar there corresponds a unique scalar (or ) such that .

(C) Multiplication is distributive with respect to addition, .

If addition and multiplication are defined within some set of objects (scalars) so that the conditions (A) , (B) , and (C) are satisfied, then that set (together with the given operations) is called a field . Thus, for example, the set of all rational numbers (with the ordinary definitions of sum and product) is a field, and the same is true of the set of all real numbers and the set of all complex numbers.

EXERCISES

Exercise 1. Almost all the laws of elementary arithmetic are consequences of the axioms defining a field. Prove, in particular, that if is a field, and if , and belong to , then the following relations hold.

.
If , then .
. (Here .)
. (For clarity or emphasis we sometimes use the dot to indicate multiplication.)
.
.
If , then either or (or both).

Exercise 2.

Is the set of all positive integers a field? (In familiar systems, such as the integers, we shall almost always use the ordinary operations of addition and multiplication. On the rare occasions when we depart from this convention, we shall give ample warning. As for "positive," by that word we mean, here and elsewhere in this book, "greater than or equal to zero." If is to be excluded, we shall say "strictly positive.")
What about the set of all integers?
Can the answers to these questions be changed by re-defining addition or multiplication (or both)?

Exercise 3. Let be an integer, , and let be the set of all positive integers less than , . If and are in , let be the least positive remainder obtained by dividing the (ordinary) sum of and by , and, similarly, let be the least positive remainder obtained by dividing the (ordinary) product of and by . (Example: if , then and .)

Prove that is a field if and only if is a prime.
What is in ?
What is in ?

Exercise 4. The example of (where is a prime) shows that not quite all the laws of elementary arithmetic hold in fields; in , for instance, . Prove that if is a field, then either the result of repeatedly adding to itself is always different from , or else the first time that it is equal to occurs when the number of summands is a prime. (The characteristic of the field is defined to be in the first case and the crucial prime in the second.)

Exercise 5. Let be the set of all real numbers of the form , where and are rational.

Is a field?
What if and are required to be integers?

Exercise 6.

Does the set of all polynomials with integer coefficients form a field?
What if the coefficients are allowed to be real numbers?

Exercise 7. Let be the set of all (ordered) pairs of real numbers.

If addition and multiplication are defined by and does become a field?
If addition and multiplication are defined by and is a field then?
What happens (in both the preceding cases) if we consider ordered pairs of complex numbers instead?

EXERCISES

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