In the next segment of the course, we will study explicitly constructed codes, and develop error-correction algorithms for them. Prominent among these will be certain algebraic constructions of codes based on polynomials over finite fields.

It is possible to get quite far treating finite fields as “black-boxes” that allow the field operations to be performed efficiently as atomic steps, along with just one important mantra:

*A non-zero polynomial of degree with coefficients from a field has at most roots in .*

But it is nevertheless desirable to have a good working knowledge of the basics of the theory of finite fields, and we will appeal to some of these results later on for list decoding some powerful algebraic codes. You are likely already familiar with this material from your undergraduate algebra. You can refer to your favorite algebra text for the basic theorems and their proofs, but I wanted to point to some notes that you can turn to if you need a refresher and a convenient reference.

So here are some excellently done notes on finite fields, written by G. David Forney and available on MIT’s OpenCourseWare for the course 6.451 Principles of Digital Communication II. These notes rigorously prove everything that we would need (and more!) from first principles, in a nice sequence.

Collected below are some basic results about finite fields, for quick reference. (I do not recall the definition of fields and the field axioms here.) All these facts are proved in the above linked notes.

- For every prime , there is a unique finite field of size that is isomorphic to which is the set under mod- addition and multiplication.
- For each prime , positive integer , and polynomial with coefficients in of degree that is
*irreducible*(in ), the set of polynomials in of degree at most with addition and multiplication of the polynomials defined modulo is a finite field (denoted ) with elements. - Every finite field is isomorphic to such a field, and therefore must have elements for some prime and positive integer .
- For every prime and integer , there exists an irreducible polynomial of degree . Therefore, there is a finite field with elements for every prime and positive integer .
- Additively, a finite field with elements has the structure of a vector space of dimension over .
- The multiplicative group of a finite field (consisting of its non-zero elements) is cyclic. In other words, the non-zero elements of a field can be written as for some .
- A with such a property is called a
*primitive element*of the field . - A field has primitive elements, where is the Euler’s totient function.

- A with such a property is called a
- All fields of size are isomorphic to for an arbitrary choice of degree irreducible polynomial .

The finite field with elements is therefore unique up to isomorphism field and will be denoted by .**Remark:**While one can pick any irreducible to represent the field as , sometimes a special choice can be judicious. For example, the complexity of multiplication is better if is sparse (i.e., has very few non-zero coefficients). - The elements of are the distinct roots of the polynomial .
- For each dividing , the field has a unique subfield of size , which consists of the roots of the polynomial .
- The polynomial is the product of all monic irreducible polynomials in whose degree divides , with no repetitions.

[…] in class (and much more). A bit of warning: I have not read the notes in any details but they are highly recommended by someone I trust. If you are not familiar with finite fields, I highly recommend that you read […]

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