Please take a look at pages 80-85 (Section 2.6 Finite fields) of this handbook, Handbook of Applied Cryptography.
I am trying to learn the mathematics enumerated in these pages. I do not need the algorithms. I will also skip any proof of theorems. The learning is for background reading as preparation for the field of cryptography that use them.
While the handbook is an excellent one and authoritative, it is somewhat concise for my purpose. It states the theories and facts without explaining the intuition behind one. Also, it does not contain many examples. This is quite natural, this is a handbook, not a textbook. I am looking for one or two texts or lecture notes dealing more or less in these areas but with elaborations and lots of examples to facilitate self-learning.
Page 86 of the same handbook suggests four references ([646], [764], [830], [841]) for further reading. I have taken a look at all of these except the first one. But these are too elaborate for the time I can spend on these topics.
I have done a thorough search over the Internet, without much success. Perhaps lack of knowledge on my part is also working as an impediment.
Your suggestions will be appreciated.
Virtually any book on algebra will contain more, but since you are especially interested in coding theory, I can recommend this coding theory book which is a little less terse (see pages 111-115).
Here's another link do a document with worked out examples. It looks like it is written for engineers (but I'm not sure if that's a plus or minus for you).
Wolfram has a short article with a worked example. In this case, Wikipedia's article might be fairly helpful, because it appears to contain a lot of examples! Be on your guard for typos, of course.
Searching "finite field pdf" turned up lots of relevant hits, including this slideshow on the topic.
The main theory of finite fields is very simple. A finite field has prime characteristic $p$ for some $p$, and then it must be of order $p^k$ for some positive integer $k$. There are finite fields of all possible orders $p^k$ for each combination of prime $p$ and positive integer $k$.
Finally, the group of nonzero elements in the field must be cyclic, and so there are things called primitive elements which generate that group.
Constructing the fields of prime order is easy: they are exactly $\Bbb F_p=\Bbb Z/p\Bbb Z$. The link I provided above furnishes the last task by giving some worked out examples of how the rest of the fields can be constructed with a quotient $\Bbb F_{p^k}=\Bbb F_p[x]/(g(x))$ where $g(x)$ is an irreducible polynomial over $\Bbb F_p$ of degree $k$.
One more thing that comes to mind for coding theory is the "Freshman's Dream Theorem" that for any polynomial $g(x)$ over $\Bbb F_{p^k}$, $g(x^p)=(g(x))^p$ which sometimes comes in handy. For example, $(x+2)^3=x^3+2^3=x+2$ over the field $\Bbb F_3$.
There are probably some finer details about fields that I'm not mentioning, but the above facts were sufficient for a few years of graduate study on coding theory in my case, anyhow.