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I know that when $K$ is a field (or even more generally a ring) then $K[x]$, the set of all the polynomials of one variable $x$ whose coefficients are in $K$, is a ring itself (with sum and product "properly" defined, not relevant for my question I think).

Now, I want to consider the particular case in which $K = \mathbb F_q$ and I know that if $q=p^r$ with p a prime and $r>1$ then $$\mathbb F_q = \left( \mathbb Z/_p[x] \right)/_{\left\langle f \right\rangle}$$ Where $f$ is an irreducible monic polynomial of degree $r$ in $\mathbb Z/_p[x]$. Moreover I know that $\mathbb F_q$ has dimention $r$ as a vector subspace on $\mathbb F_p$ so I can represent each element in $\mathbb F_q$ as a tuple $(a_0,a_1,\dots, a_{r-1})\in\mathbb Z/_p^{\:r}$.

Now with that being said, how do I represent an element in $\mathbb F_q[x]$?

Baffo rasta
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  • Not a very clear question. The elements $F_q[x]$ are polynomials of the form $a_0+a_1x+\ldots+a_sx^s$, $a_i\in F_q$. I think that's how you should think of them. – kabenyuk Sep 08 '21 at 15:49
  • @kabenyuk: OK I know that, but what are $a_0,a_1,\dots, a_s$? Are those polynomials too since $\mathbb F_q$ is defined as above? – Baffo rasta Sep 08 '21 at 15:53
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    The field $F_q$ can be constructed in different ways. In order to work with polynomials, it doesn't matter which way the field is constructed. Or formulate your question more precisely. – kabenyuk Sep 08 '21 at 16:26
  • @kabenyuk: I think my question is clear enough: all I'd like to know is what kind of mathematical objects are $a_0,a_1,\dots$, whether they're polynomials, tuples, numbers, and what possible values can they take, that's all. – Baffo rasta Sep 08 '21 at 16:31
  • These objects are just elements of $F_q$ and nothing more. They are neither polynomials, nor tuples, nor numbers, and so on. – kabenyuk Sep 08 '21 at 16:42
  • @Kabenyuk: OK then let's make this more practical. Let $\mathbb F_9[x] = \mathbb F_3[x]/_{\left\langle x^2+x+2\right\rangle}$, can you please list a non-null element of $\mathbb F_9[x]$? Appreciated. – Baffo rasta Sep 08 '21 at 16:43
  • @kabenyuk I think OP's question is a good one. They are asking how one represents elements of finite fields. It's fine abstractly to say that an element of $\mathbb F_9$ is just an element of $\mathbb F_9$, but concrete representations help too. A complex number is just an element of the algebraic closure of the reals, but isn't it nice to be able to actually write one down? To OP, a related question is how computers represent elements of finite fields. I believe in computer science they often refer to these as "Galois fields," so you could begin your investigation there. – paul blart math cop Sep 08 '21 at 21:40
  • @paulblartmathcop: My current best guess is that given a primitive element $\omega$ of $\mathbb F_q$ then each element of $\mathbb F_q[x]$ should be represented as $a_0 + a_1x+\dots+a_sx^s$ as Kabenyuk was saying, where $a_0,a_1\dots$ all are of the form of $\omega^r$ for some $r=1,\dots, q-1$ or $0_{\mathbb F_q}$, am I wrong? – Baffo rasta Sep 08 '21 at 21:46
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    As I understand it, there are many ways you can represent elements of finite fields. Your way certainly works. You could also find a generating element $\alpha$ (and your primitive element $\omega$ will do) and express elements of $\mathbb F_9$ as $a + b \alpha$ for $a, b \in \mathbb F_3$ unique. This can be stored as a tuple. I think (but I am far from an expert) that computer scientists like this second representation for $\mathbb F_{2^n}$, as then its elements can be written explicitly as an $n$ bit string. Once we represent finite fields, their polynomials can be represented as tuples. – paul blart math cop Sep 08 '21 at 21:53
  • This question (and its cousins crop up occasionally). A bit of discussion here. They don't address quite the concern here. I spot a confusion above (common for students of this topic, trust me!) : If you write $\Bbb{F}_9$ as $\Bbb{F}_3[x]/(x^2+x+2)$ then that ties $x$, and to discuss polynomials with coefficients in $\Bbb{F}_9$ we need to use another variable. I usually don't want to go that way. Rather I introduce the notation $\alpha$ for the coset $x+(x^2+x+2)$ in the quotient ring $\Bbb{F}_3[x]/(x^2+x+2)$. – Jyrki Lahtonen Sep 10 '21 at 20:30
  • (cont'd) From that point on $\alpha$ becomes just some obscure quantity that satisfies the equation $\alpha^2+\alpha+2=0$. Then we can write $$\Bbb{F}_9=\Bbb{F}_3(\alpha)={a+b\alpha\mid a,b\in\Bbb{F}_3}.$$ Using $\alpha$ frees up $x$. And it becomes unproblematic to discuss the polynomials $\Bbb{F}_9[x]$. They are objects like $$(1+2\alpha)+\alpha x+ (2+\alpha) x^2+2x^3.$$ The point in $\Bbb{F}_9[x]$ is that the $x$ is totally unrelated to the $x$ in $\Bbb{F}_3[x]/(x^2+x+2)$. – Jyrki Lahtonen Sep 10 '21 at 20:34
  • (cont'd) This is similar to the construction of the field of complecx numbers as the quotient ring $\Bbb{C}=\Bbb{R}[x]/(x^2+1)$, when we similarly decide to rename the coset $x+(x^2+1)$ and decide to denote it $i$ instead. It is just some obscure quantity that satisfies the equation $i^2+1=0$. We then have no problems dealing with the polynomials from $\Bbb{C}[x]$ such as $1+2x+(3+i)x^2-ix^3$. – Jyrki Lahtonen Sep 10 '21 at 20:37

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