Math 360, Fall 2013, Assignment 12

Algebra begins with the unknown and ends with the unknowable.

- Anon.

Carefully define the following terms, then give one example and one non-example of each:

1. Zero-divisor.
2. (Integral) domain.
3. Characteristic (of a ring).

Carefully state the following theorems (you need not prove them):

1. Cancellation laws (for an integral domain).
2. Relationship between fields and integral domains (i.e. which fields must be domains, and which domains must be fields).
3. Little Theorem of Fermat.
4. Euler's Theorem.

Solve the following problems:

1. Section 19, problems 1, 5, 7, 9, and 23.
2. Section 20, problems 5, 9, and 10.

--------------------End of assignment--------------------

Solutions:

Definitions:

1. Zero Divisor

   Let \(R\) be a ring and \(a\) and \(b\) two non-zero element of \(R\). \(a\) and \(b\) are zero-divisors if:$$ a*b = 0 $$

   Example:

   2 and 4 are zero-divisors in \(\mathbb{Z}_8\), because \(4*2=0\) mod 8.

   Non-Example:

   There are no zero divisors in \(\mathbb{Z}\).

2. Integral Domain

   An integral domain is a ring with no zero divisors. So \(R\) is an integral domain if there are no non-zero elements \(a\) and \(b\) such that:$$ a*b=0 $$

   Example:

   The integers are an integral domain. There is no way to multiply non-zero integers and get zero.

   Non-Example:

   \(\mathbb{Z}_8\) is not an integral domain (see example above). More generally, any \(\mathbb{Z}_n\) where \(n\) is composite is not an integral domain.

3. Characteristic of a Ring

   Consider a unital ring \(R\), and a function \(\phi:\mathbb{Z}\rightarrow R\) such that:$$ \phi(n) = \sum_0^n1_R $$

   The characteristic of \(R\) is the smallest positive \(n\) such that \(\phi(n)=0_R\). If there is no such \(n\), then the characteristic of \(R\) is 0. More intuitively, the characteristic of \(R\) is the minimum number of times you need to add the multiplicative identity (of \(R\)) to itself to get 0.

   Example:

   The characteristic of \(\mathbb{Z}_n\) is \(n\). Obviously if you add 1 to itself \(n\) times you get \(n\), which is 0 in \(\mathbb{Z}_n\).

   Non-Example:

Theorems:

1. Cancellation Laws for Integral Domains

   In an integral domain \(D\), if we have:$$ ab = ac $$

   then \(b=c\). (since integral

2. Relationship between Fields and Integral Domains

   A finite integral domain is always a field. A subring of a field is always an integral domain.

3. Little Theorem of Fermat

   If \(p\) is prime, and \(a\) is not a multiple of \(p\), then:$$ a^{p-1}\equiv 1 \mod{p} $$

4. Euler's Theorem

   Let \(a\) and \(n\) be relatively prime integers. Then:$$ a^{\varphi(n)}\equiv 1 \mod{n} $$

   where \(\varphi\) is Euler's totient function (i.e. the number of integers less than \(n\) that are relatively prime with \(n\).

Book Problems:

1. 19.1

   First, factorize the polynomial to get:$$ x(x-3)(x+1)=0 $$

   Now we need to list the pairs of zero divisors of the ring we're working in:$$ 0,x\\ 3,4\\ 2,6\\ $$

   Since \(\mathbb{Z}_{12}\) is commutative, we don't need to flip the pairs. Continuing on, we find that \(x\) can be:$$ 0,3,11,5,8 $$

   I don't see a fast, general algorithm for solving this problem - it's easy to find some roots, but not all the roots.

2. 19.5

   The characteristic of \(2\mathbb{Z}\) is 0.

3. 19.7

   The characteristic of \(\mathbb{Z}_3\times 3\mathbb{Z}\) is 0.

4. 19.9

   The characteristic of \(\mathbb{Z}_3\times \mathbb{Z}_4\) is 12.

5. 19.23

   First, 1 and 0 are idempotent. Let \(b\) be an idempotent value not equal to 0 or 1. Then:$$ b^2 = b\\ b(b) = 1*b\\ b = 1 $$

   We can use cancellation because \(b\) is required to not be a zero divisor. (Cancellation is allowed as long as nothing involved is a zero divisor.)

6. 20.5

   $$ 37^{49} \mod{7} \equiv 2^{49}\mod{7} \equiv 2^{6^8}*2\mod{7} \equiv 2 \mod{7} $$

7. 20.9

   The totient function of \(pq\) (both prime) is \((p-1)(q-1)\).

8. 20.10

   $$ 7^{1000}\mod{24}\\ \phi(24) = 7\\ 7^{7} \equiv 1 \mod{24}\\ 7^{1000} = 7^{7^142}*7^6\\ 7^{1000} \equiv (7^7)^{142}*7^6\mod{24}\\ \equiv 7^6\mod 24\\ \equiv 1\mod{24} $$