2.59 Theorem (Cancellation laws.) Let be a field, let be elements in , and let . Then

(2.60) (2.61) (2.62) (2.63)

( 2.60 ) and ( 2.61 ) are called cancellation laws for addition, and ( 2.62 ) and ( 2.63 ) are called cancellation laws for multiplication.

Proof: All of these results are special cases of the cancellation law for an associative operation (theorem 2.19).

2.64 Theorem. In any field





Proof: These are special cases of the remark made earlier that an identity element is always invertible, and is its own inverse.

2.65 Theorem (Double inverse theorem.) In any field ,







Proof: These are special cases of theorem 2.17.



I will now start the practice of calling a field . If I say `` let be a field" I assume that the operations are denoted by and .

2.66 Theorem. be a field. Then

Letbe a field. Then



Proof: We know that , and hence



2.67 Corollary. Let be a field. Then for all , .

2.68 Theorem. Let be a field. Then for all in

(2.69)



Proof:

Case 1: Suppose . Then (2.69) is true because every statement implies a true statement. Case 2: Suppose . By theorem 2.66, , so





2.70 Remark. We can combine theorem ,

We can combine theorem 2.66 , corollary 2.67 and theorem 2.68 into the statement: In any field



2.71 Exercise. Let be a field. Prove that has no multiplicative inverse in . Letbe a field. Prove thathas no multiplicative inverse in

2.72 Theorem (Commutativity of addition.) Let be any field. Then is a commutative operation on .

Proof: Let be elements in . Then since multiplication is commutative, we have



2.73 Remark. Let be a field, and let . Then

(2.74)

Letbe a field, and let. Then

(2.75)



Proof:





2.76 Theorem. Let be a field. Then





Proof: Let . By (2.74) it is sufficient to prove







2.77 Exercise. be a field, and let . Prove that a) b) c) Letbe a field, and let. Prove that

2.78 Exercise. be a field and let be non-zero elements in . Prove that

A Letbe a field and letbe non-zero elements in. Prove that



2.79 Definition (Digits.) be a field. We define

Letbe a field. We define

I'll call the set

I'll call the set

the set of digits in . If are digits, I define

(2.80)

the set of. Ifare digits, I define and

(2.81)

and Here should not be confused with . Hereshould not be confused with

2.82 Example.





In general, if , I define

In general, if, I define

Then for all digits Then for all digits





so, for example

so, for example



2.83 Remark. The set of digits in may contain fewer than ten elements. For example, in , The setof digits inmay contain fewer than ten elements. For example, in





and you can see that . and you can see that

2.84 Theorem. In any field , and .



Proof:



2.85 Exercise. Prove that in any field , and . Prove that in any fieldand

2.86 Exercise. Prove that in any field , . Prove that in any field

2.87 Remark. After doing the previous two exercises, you should believe that the multiplication and addition tables that you learned in elementary school are all theorems that hold in any field, and you should feel free to use them in any field.