got this assignment from my coding class and don't know if I've made it correct. Can someone tell if my methods for solving the tasks are correct?

Let $f(x) = 1 + x ^3 + x ^4$ . It is given that $f(x)$ is irreducible in $K[x]$. Let $F = GF(2^4 )$ be $K[x] modulo f(x)$ and let be $x modulo f(x)$ in $F$.

a) By making a table expressing $1, \beta, \beta^2... \beta^{14}$ in the form $a_0 + a_1\beta + a_2\beta^2 + a_3\beta^3$ (or $a_0a_1a_2a_3)$ with the a_i in $K$, verify that $\beta$ is a primitive element of $F$.

So after constructing the table I made the following equation:

$p(\beta)=1+\beta^3+\beta^4$ and after inputting the corresponding values from the table $p( \beta) = 1000 + 0001 + 1001 = 0000$ so $\beta$ is a primitive

b) Let $\alpha$ be $\beta^{12}$. Is $\alpha$ a primitive element of F?

so the equation is now $p(\beta^{12})=1+(\beta^{12})^3+(\beta^{12})^4 = 1 + \beta^{36} + \beta^{48}$. Here I'm not actually sure. The table is only till $\beta^{14}$ and in the book i've found that $\beta^{15} = 1$? so following this $\beta^{36} = \beta^6$ and $\beta^{48} = \beta^3$ and the equation gets the following form $p(\beta^{12})=1000+1111+0001=0110$ so $\beta^{12}$ is not a primitive?

c) Find the minimal polynomial $m(x)$ in $K[x]$ for $\alpha$ as in (b)

so $\alpha = \beta^{12}$, and $m_a(\beta^{12})=a_01+a_1\beta^{12}+a_2(\beta^{12})^2+a_3(\beta^{12})^3+a_4(\beta^{12})^4=a_01+a_1\beta^{12}+a_2\beta^{24}+a_3\beta^{36}+a_4\beta^{48} = a_0(1000)+a_1(1100)+a_2(1010)+a_3(1111)+a_4(0001)$. solving the equation gives $a_0=a_1=a_2=a_3=a_4=1$ and $m_\alpha(x) = 1 + x + x^2 + x^3 + x^4$ and the roots are $\{\beta^{12},\beta^{24},\beta^{36},\beta^{48}\}$ and $m_{12}(x)=m_{24}(x)=m_{36}(x)=m_{48}(x)$ denote the minimal polynomials of $\beta^i$


Your test that $1000+0001+1001 = 0000$, that is, $1+\beta^3+\beta^4 = 0$ shows only that $\beta$ is a root of $1+x^3+x^4$ and not that $\beta$ is primitive (meaning of order $15$). What shows that $\beta$ is primitive is that $1, \beta, \beta^2, \ldots, \beta^{14}$ all have different representations as polynomials of degree at most $3$ in $\beta$ and so $\beta$ has order $15$ or more; and when you compute $\beta^{15}$ as a polynomial of degree at most $3$ in $\beta$, you find that $\beta^{15} = 1$ and so $\beta$ is indeed of order $15$ and hence primitive.

On the other hand, $\alpha = \beta^{12}$ is not a primitive element. Here, it is simpler to use the result that

$${\sf ord}(\beta^k) = \frac{{\sf ord}(\beta)}{\gcd({\sf ord}(\beta), k)}$$ to deduce that $\beta^{12}$ is an element of order $5$. Or, you could try showing that $\alpha^5 = \beta^{60} = 1$, hopefully without grinding out the table to $60$ terms (hint: you already know that $\beta^{15}=1$). So, $\alpha$ is an element of order $5$ or a divisor of $5\quad$....

  • $\begingroup$ Thank You very much! I see I've understood the stuff from the book not correct! so summing up, looking for the roots doesn't tell anything about the primitive. (c) correct luckily :) $\endgroup$ – mjanisz1 Oct 7 '13 at 22:19
  • $\begingroup$ Sorry but reading this now i totally don't understand your solution? I assume primitive is of order 15 because the field is $2^4$ so $16-1 = 15$ How to check the order of other polynomials in that field? $\endgroup$ – mjanisz1 Oct 8 '13 at 18:34
  • $\begingroup$ What is meant by polynomials in that field? The irreducible polynomial divisors of $x^{15}-1$ are $$x+1, x^2+x+1, x^4+x^3+x^3+x^2+x+1,x^4+x^3+1, x^4+x+1$$ if which you know already that the roots of $x^4+x^3+1$ are of order $15$ and those of $x^4+x^3+x^2+x+1$ of order $5$. Since ${\sf{ord}}(\beta^{-1}) = {\sf{ord}}(\beta^{14}) = 15$, you can show that $x^4+x+1$ is also a primitive polynomial. All the remains is the orders of the roots of $x+1$ and $x^2+x+1$. Make a list of all elements whose orders you have found to figure out which elements are candidates for roots of these polynomials. $\endgroup$ – Dilip Sarwate Oct 8 '13 at 19:04
  • $\begingroup$ Sorry, I still don't understand Your method of showing how its primitive. I've found in my book that $βi ∈ GF(2^r)$ is primitive iff $gcd(i, 2^r – 1) = 1$ but don't know how to prove it. $\endgroup$ – mjanisz1 Oct 8 '13 at 19:18
  • $\begingroup$ What do you want to prove? The statement that if $\beta$ is a primitive element of $\text{GF}(2^r)$, then $\beta^i$ is a primitive element iff $\gcd(i,2^r-1)$? Note that your statement of this result is not true since you do not restrict $\beta$ to be a primitive element. As to "my" method, I merely said that if $1, \beta, \beta^2, \ldots, \beta^{14}$ are all different elements of the field, and $\beta^{15} = 1$, then $\beta$ is an element of order $15$ (this is the definition, there is nothing to prove here) and since $\beta\in\text{GF}(2^4)$, it is a primitive element $\endgroup$ – Dilip Sarwate Oct 8 '13 at 22:54

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