Consider a field $F_{16} = F_{2}(a)$ where minimal polynomial of element $a$ is $m_{a,F_{2}} = a^4+a+1$. How many elements are in the field: $$F_{2}(a^2+a)$$

My idea was to find minimal polynomial of $a^2+a$ in $F_{16}$ which I understand as $F_2/(a^4+a+1)$ - meaning operations mod $a^4+a+1$. That gave me $m_{a^2+a,F_{16}} = x^2+x+1$. And than use relation between degree of a minimal polynomial and dimension of an extension of a fild.

But that does not seem right. Any ideas how to start?

  • $\begingroup$ You can find related things in a recent question : (math.stackexchange.com/q/2297568) $\endgroup$ – Jean Marie Jun 9 '17 at 9:24
  • $\begingroup$ Note that we clearly have $F_2(a^2+a)\subseteq F_{16}=F_2(a)$, so the answer is at most $16$. $\endgroup$ – Arthur Jun 9 '17 at 9:27
  • $\begingroup$ The minimal polynomial of $a^2+a$ over $F_{16}$ is $x-(a^2+a)$, which is of first degree. $\endgroup$ – Arthur Jun 9 '17 at 11:08

Step one: Don't write the minimal polynomial of $a^2+a$ in terms of $a$. The minimal polynomial of $a^2+a$ is $x^2+x+1$, as we have that $$(a^2+a)^2 + (a^2+a)+1 = 0.$$ If you use $a$ as the variable here, things get really confusing. Next, this is the minimal polynomial over $\mathbb{F}_2$, not over $\mathbb{F}_{16}$. As this is a polynomial of degree two, you should get some ideas on how big the field is.

  • $\begingroup$ Why is it minimal polynomial over $F_2$? Because $(a^2+a)^2 + (a^2+a) + 1 = a^4+2a^3+a^2+a^2+1=a^4+a+1$, and that is not zero in $F_2$ (the residual after division by $m_a,F_2$ is). $\endgroup$ – Rosta655321 Jun 9 '17 at 9:42
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    $\begingroup$ @Rosta655321: I'm sorry, but $a^4+a+1$ is $0$, by hypothesis. The $0$ element is the same in $\mathbf F_2$ and in $\mathbf F_{16}$. $\endgroup$ – Bernard Jun 9 '17 at 10:21
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    $\begingroup$ Maybe try to write your minimal polynomials really as polynomials, not as a relation fulfilled by the element, that might help. $\endgroup$ – Dirk Jun 9 '17 at 10:22

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