We continue from last lecture. Examination of a few particular cases finally allows us to complete Example 6. The answer to whether (with the operations and constants defined last time) is a field splits into three parts. The first is straightforward.
Lemma 10. For all positive integers satisfies all the properties of fields except possible the existence of multiplicative inverses. .
This reduces the question of whether is a field to the problem of finding multiplicative inverses, which turns out to be related to properties of
Lemma 11. If is not prime, then is not a field.
Proof. If is not prime, then either and has only one element, but in fields or else and there is some such that and But then has no multiplicative inverse. Otherwise, for some we would have that i.e., there is some such that But then and since divides both and then also divides 1, contradiction.
Lemma 12. If is prime, then is a field.
Proof. Suppose that Then so so there are integers such that But then
The following observations are useful when studying fields:
- No field has zero divisors, i.e., if is a field, and then either or
This is because if then it has a multiplicative inverse . Multiplying (on the left) both sides of by we get
- In a field there is only one additive identity and one multiplicative identity.
If and let be a multiplicative inverse of Then The argument for additive identities is similar.
- In a field there is only one additive inverse for any element and one multiplicative inverse for any nonzero element.
Suppose that Then The argument for multiplicative inverses is similar.
Due to these observations, we just use familiar notation and, for example, write for the additive inverse of and or for the multiplicative inverse of
- has zero divisors when it is not a field (and ).
This is because, by the lemmas above, if is not a field then is not prime, so we can find such that and therefore and are zero divisors.
On the other hand, there are structures that satisfy all properties of fields except the existence of multiplicative inverses, and yet they have no zero divisors. For example, consider Let’s now try to continue our list of examples.
Example. 7. Are the with prime the only finite fields? For example, can we exhibit a field with 4 elements? To answer this question, let’s first see what additional properties such a field would have to satisfy. Next lecture we will prove the following:
Lemma 13. Suppose that is a finite field. Then there is some natural number such that the sum of ones vanishes, The least such is a prime that divides the size of the field.