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<a<n and a|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[0]_p. Then p\not|a, so (a,p)=1, so there are integers b,c such that ab+pc=1. But then {}[a]_p[b]_p=[1]_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<a,b<n 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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One Response to 305 -Fields (2)

  1. […] -Fields (3) At the end of last lecture we arrived at the question of whether every finite field is a for some […]

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