i just started studying coding theory from the book "error control coding" by lin and costello. In the 2nd chapter he has given a brief overview of the galois fields. No there are a few confusion that i have and i would be thankful to whoever can help me calrify them.
1. While defining a field (or a group) it sometimes seems to me that you can define addition and multiplication whatever way u want it . like sometimes its modulo m where m is the size of the finite field or group sometimes its polynomial addition etc. so please kindly can somebody explain what are the restriction on how we define these binary operators.
2. Now it was mentioned in the book that galois fields (considering modulo addition and modulo multiplication) exist for size p^k where p is a prime and k is a
positive integer. Why?. i tried to construct for a couple of non primes but the table for multiplication doesn't come out right.
3. What is the Significance of the characteristic of a field.
4. I have read two different definitions of irreducible polynomials over GF(q). One says that a polynomial is irreducible over GF(q) if it doesn't have any root in GF(q). Second one which i read in the above mentioned book and which I am not sure if it holds in general of only for GF(2^k) says that a polynomial of degree n over Gf(q) is irreducible if no polynomial over GF(q) of degree less than n and greater than 0 is a factor of it. So are the two equivalent if yes how and if no so whcih one is right or more general.
5.Can somebody explain to me comprehensively the procedure of extending a finite field. I have an idea abt it but there are a lot of confusions so rather than posting all of them may be a brief idea by someone can help me better.
6. if i use different irreducible polynomials of the same degree m to extend a field GF(p) i will get different fields of the same size??
thanks in anticipation