## 305 -Fields (2)

We continue from last lecture. Examination of a few particular cases finally allows us to complete Example 6. The answer to whether ${\mathbb Z}_n$ (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 $n>1,$ ${\mathbb Z}_n$ satisfies all the properties of fields except possible the existence of multiplicative inverses. $\Box$.

This reduces the question of whether ${\mathbb Z}_n$ is a field to the problem of finding multiplicative inverses, which turns out to be related to properties of $n.$

Lemma 11. If $n$ is not prime, then ${\mathbb Z}_n$ is not a field.

Proof. If $n$ is not prime, then either $n=1$ and ${\mathbb Z}_1$ has only one element, but in fields $0\ne1,$ or else $n>1$ and there is some $a$ such that $1 and $a\mid n.$ But then ${}[a]_n$ has no multiplicative inverse. Otherwise, for some $k,$ we would have that $ak\equiv1\mod n,$ i.e., there is some $m$ such that $ak-1=nm.$ But then $1=ak+nm$ and since $a$ divides both $ak$ and $nm,$ then $a$ also divides 1, contradiction. ${\sf QED}$

Lemma 12. If $p$ is prime, then ${\mathbb Z}_p$ is a field.

Proof. Suppose that ${}[a]_p\ne_p.$ Then $p\nmid a,$ so $(a,p)=1,$ so there are integers $b,c$ such that $ab+pc=1.$ But then ${}[a]_p[b]_p=_p.$ ${\sf QED}$

The following observations are useful when studying fields:

• No field has zero divisors, i.e., if ${\mathbb F}$ is a field, $a,b\in{\mathbb F},$ and $ab=0,$ then either $a=0$ or $b=0.$

This is because if $a\ne0$ then it has a multiplicative inverse $c$. Multiplying (on the left) both sides of $ab=0$ by $c$ we get $0=c0=c(ab)=(ca)b=1b=b.$

• In a field there is only one additive identity and one multiplicative identity.

If $z\ne0$ and $zw_1=z,$ let $v$ be a multiplicative inverse of $z.$ Then $w_1=1w_1=(vz)w_1=v(zw_1)=vz=1.$ 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 $z+w_1=0=z+w_2.$ Then $w_1=0+w_1=(z+w_2)+w_1=(w_2+z)+w_1=w_2+(z+w_1)=w_2+0=w_2.$ The argument for multiplicative inverses is similar.

Due to these observations, we just use familiar notation and, for example, write $-z$ for the additive inverse of $z$ and $z^{-1}$ or $\displaystyle \frac1z$ for the multiplicative inverse of $z\ne0.$

• ${\mathbb Z}_n$ has zero divisors when it is not a field (and $n>1$).

This is because, by the lemmas above, if ${\mathbb Z}_n$ is not a field then $n$ is not prime, so we can find $1 such that $n=ab,$ and therefore ${}[a]_n$ and ${}[b]_n$ are zero divisors.

On the other hand, there are structures $(R,+,\times,0,1)$ that satisfy all properties of fields except the existence of multiplicative inverses, and yet they have no zero divisors. For example, consider ${\mathbb Z}.$ Let’s now try to continue our list of examples.

Example. 7. Are the ${\mathbb Z}_n$ with $n$ 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 ${\mathbb F}$ is a finite field. Then there is some natural number $n>0$ such that the sum of $n$ ones vanishes, $1+\dots+1=0.$ The least such $n$ is a prime that divides the size of the field.

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