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I have a field $\mathbb F_{2^4}$, and it is represented as a residue ring of the
polynomials over $\mathbb F_2$ modulo the polynomial $\beta^4 + \beta^3 + \beta^2 + \beta + 1$.

I want to express each element in this field as a power of a primitive element $β+1$.

My questions are:

  1. What are the elements of the fields?
  2. How do I express them as powers of the given primitive element?

Any hints will be very helpful.

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For the second question: If you say $g:= \beta + 1$, and want to express them as powers of $g$, then this is just $g^1,g^2,\dots,g^{15}=g^0$ –  tylo Apr 11 at 12:23
    
@tylo : Hello... Thanks that helps... I figured out that for g^15 = g^8... Is it something weird? Am I doing something wrong or it is possible. I did not try after g15. –  kingmakerking Apr 11 at 15:09
    
Why would it be $g^8$? If your element is primitive and the polynomial is irreducible (I did not check), the multiplicative group generated by $g$ is cyclic and has exactly 15 distinct elements. –  tylo Apr 11 at 15:23

1 Answer 1

up vote 4 down vote accepted

Just enumerate all powers of your primitive element. There are only 15 of them. That will give you a lookup table that allows you to translate from field element to its expression as a power of the primitive element.

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1  
tallows->allows; Personally I would write "There are only 15 of them before you reach an expression equal to 1". –  figlesquidge Apr 11 at 15:39

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