A field F is a set that includes 0 and has two abelian operations defined, usually called multiplication and addition. F is a group under addition and F - {0} is a group under multiplication.
There are many well known infinite sets that are fields, including the real numbers ℝ, the rationals ℚ, and the complex numbers ℂ. The integers ℤ are not a field because there are no multiplicative inverses. But when we take ℤp where p is prime, this creates a finite field.
Let's look at ℤ5 - {0} under multiplication.
1 2 3 4
1| 1 2 3 4
2| 2 4 1 3
3| 3 1 4 2
4| 4 3 2 1
1 is always its own inverse, and in ℤ5 - {0}, 4 is also its own inverse. It will always be true that (n-1)²≣ 1 (mod n) for all n > 1.
Because 6 is not prime, ℤ6 - {0} under multiplication is not a group.
1 2 3 4 5
1| 1 2 3 4 5
2| 2 4 0 2 4
3| 3 0 3 0 3
4| 4 2 0 4 2
5| 5 4 3 2 1
Because both 2 and 3 divide 6, they do not have multiplicative inverses. 4 doesn't divide 6, but they have a common factor of 2. If we want to make a multiplicative group mod 6, we have to use the numbers relatively prime to 6, which are 5 and 1.
1 5
1| 1 5
5| 5 1
For any prime p, all the integers from 1 to p-1 are relatively prime, so the multiplication table will never have two non-zero numbers that multiply to 0.
There are also finite fields of the order p². In the next post, we will look at a finite field of order 4.
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