Page 192 - DMTH403_ABSTRACT_ALGEBRA

Unit 18: Integral Domains

Theorem 6: Every finite integral domain is a field. Notes
Proof: Let R = {a, = 0, a = 1, a ,....., a,] be a finite domain. Then R is commutative also. To show
2
1
that R is a field we must show that every non-zero element of R has a multiplicative inverse.
So, let a = a be a non-zero element of R (i.e., i  0). Consider the elements aa , ..., aa . For every
n
j
1
j  0, a  0; and since a  0, we get aa  0.
j
j
Hence, the set { aa , ..., aa } G (a,, ..., a,}.
n
1
Also, aa, , aa ,..., aa, are all distinct elements of the set {a,, ...., a,}, since aa = na a = a, using the
k
j
j
j
cancellation law for multiplication.
Thus, {aa , ...., aa } = [a; ,...., a }.
n
n
1
In particular, a, = aa, i.e., 1 = aa for some j. Thus, a is invertible in R. Hence every non-zero
j
j
element of R has a multiplicative inverse. Thus, M is a field.
Using this result we can now prove a theorem which generates several examples of finite fields.
Theorem 7: Z is a field if and only if n is a prime number.
n
Proof: From theorem 1 you know that Z is a domain if and only if n is a prime number. You also
n
know that Z has only n elements. Now we can apply Theorem 6 to obtain the result.
n
Theorem 7 unleashes a load of examples of fields : Z , Z , Z , Z ,, and so on. Looking at these
2
7
3
5
examples, and other examples of fields, can you say anything about the characteristic of a field?
In fact, using Theorems 4 and 5 we can say that.
Theorem 8: The characteristic of a field is either zero or n prime number.
So far the examples of finite fields that you have seen have consisted of p elements, for some
prime p. In the following exercise we give you an example of a finite field for which this is not
so.
Theorem 9: Let R be a ring with identity. Then R is a field if and only if R and {0} are the only
ideals of R.
Proof: Let us first assume that R is a field. Let I be an ideal of R. If I  {0), there exists, a non-zero
element x  I. As x  0 and R is a field, xy = 1 for some y  R. Since x  I and I is an ideal, xy  I,
i.e., 1  I.
Conversely, assume that R and { 0 } are the only ideals of R. Now, let a  0 be an element of R.
Then you know that the set Ra = [ra | r  R] is a non-zero ideal of R. Therefore, Ra = R.
Now, 1  R = Ra. Therefore, 1 = ba for some b  R, i.e., a exists. Thus, every non-zero element
-1
of R has a multiplicative inverse. Therefore, R is a field.
Using Theorem 9, we can obtain some interesting facts about field homomorphisms (i.e., ring
homomorphisms from one field to another). We give them to you in the form of an exercise.
Now that we have discussed domains and fields, let us look at certain ideals of a ring, with
respect to which the quotient rings are domains or fields.
Self Assessment

1. Several authors often shorten the term .................... to domain.
(a) integral domain (b) abstract domain
(c) different domain (d) prime domain

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