Find all irreducible polynomials of degree 5 with coefficients in GF(2). (Hint 1: list all of the possible polynomials first. Then you can easily deduce that many of these are reducible.) (Hint 2: For the remaining possible polynomials consider all irreducible polynomials of degree <5 and check for factors) Any help and good explainations would be apprecated. Thank you.
Find all irreducible polynomials of degree 5 with coefficients in GF(2). (Hint 1: list all of the possible polynomials first. Then you can easily deduce that many of these are reducible.) (Hint 2: For the remaining possible polynomials consider all irreducible polynomials of degree <5 and check for factors) Any help and good explainations would be apprecated. Thank you.
Sol:-
There are a total of 25 = 32 possible polynomials of degree 5 with coefficients in GF(2), since each coefficient can be either 0 or 1. However, many of these are reducible, so we need to identify the irreducible ones.
One way to do this is to use the fact that any irreducible polynomial of degree n in GF(2) is a factor of x^{2^n} - x in GF(2). This is known as the "irreducibility test" for GF(2) polynomials.
Using this test, we can eliminate many of the possible polynomials. For example, we know that x^5 - x is not irreducible, because it has x as a factor. Similarly, we can check that x^5 + x, x^5 + x^2, and x^5 + x^3 are all reducible by factoring them into irreducible polynomials of lower degree.
This leaves us with the following possible irreducible polynomials of degree 5 in GF(2):
x^5 + x^4 + x^2 + x + 1
x^5 + x^4 + x^3 + x + 1
x^5 + x^4 + x^3 + x^2 + 1
x^5 + x^4 + x^3 + x^2 + x + 1
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