How Do You Compute Minimal and Characteristic Polynomials in F16 Over F2?

[sara15]

Hey guys,

I really need some help please!
I would really appreciate it if anyone can help out,

if we have F16 = F2/(x^4+x+1). can anyone explain to me how can I compute the minimal polynomials and the characteristic polynomils over F2 of elements of F16 and to point out the primitive ones . I have difficulty to understand this question.
Thanks

 

[micromass]

OK, here's what you do to find the minimal polynomial. Take an element a in F16. First you need to find a polynomial such that a is a root of the polynomial, then you need to make sure that this polynomial is minimal.

Let me give an example: take x in F16 (where we interpret x as a polynomial).
Since F16=F2[X]/(X^4+X+1), we see that x is a root of the polynomial X^4+X+1. Furthermore, this is the minimal polynomial, since X^4+X+1 is irreducible over F2.

This is how you have to do these kind of things.
By the way, could you say what you mean with characteristic and primitive polynomials? I only know these terms with respect to linear algebra...

 

[sara15]

OK, here's what you do to find the minimal polynomial. Take an element a in F16. First you need to find a polynomial such that a is a root of the polynomial, then you need to make sure that this polynomial is minimal.

Let me give an example: take x in F16 (where we interpret x as a polynomial).
Since F16=F2[X]/(X^4+X+1), we see that x is a root of the polynomial X^4+X+1. Furthermore, this is the minimal polynomial, since X^4+X+1 is irreducible over F2.

This is how you have to do these kind of things.
By the way, could you say what you mean with characteristic and primitive polynomials? I only know these terms with respect to linear algebra...

Thanks for replying to my question. I do not know how to find the characteristic polynomial
by this way (x-alpha)(x-sigma(alpha))...(x-sigma^n-1(alpha)) this is called the characteristic polynomial of alpha over Fq , where alpha is in Fq^n.
and the primitive polynomial is a monic polynomial of degree n over Fq and has a primitive element of Fq^n as one of its roots. The primitive element is an element that has order q-1