Math 360, Fall 2014, Assignment 13

Algebra begins with the unknown and ends with the unknowable.

- Anonymous

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

1. Group of units (of a unital ring).
2. Euler $\phi$-function.
3. Field of fractions (of an integral domain).

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

1. Chinese remainder theorem at the level of rings.
2. Theorem describing the group of units of a product of two unital rings.
3. Formula describing $\phi(nm)$ when $n$ and $m$ are relatively prime.
4. Formula for $\phi(p^n)$ when $p$ is prime.
5. Euler's Theorem.
6. Fermat's Theorem.
7. Theorem describing which rings can be embedded in fields.

Solve the following problems:

1. Section 20, problems 5, 7, and 10.
2. Section 21, problem 1.

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

Questions:

Does anyone know the theorem number of the chinese remainder theorem in the book? The book doesn't refer to it as the chinese remainder theorem.

I don't think the book even states the CRT at ring level. Here it is: $\mathbb{Z}_n\times\mathbb{Z}_m$ is isomorphic (as a ring) to $\mathbb{Z}_{nm}$ if and only if $n$ and $m$ are relatively prime. - Steven.Jackson (talk) 12:40, 7 December 2014 (UTC)