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Let $f(t)$ be a primitive polynomial in a finite field, of degree $r$, ${F_q}$ (that is, an irreducible polynomial whose roots in a splitting field have order $q^r-1$). This polynomial can also be viewed as the characteristic polynomial of a linear recurrence. It generates two periodic sequences in $F_q$: the null sequence $(0,0,...)$ and a sequence ("unit sequence" or "impulse-response sequence", as in "Introduction to Finite Fields" of Lidl & Niederreiter) with lenght $q^r-1$. As an example $t^2-t-1=t^2+2t+2$ is primitive in $F_3[t]$. It is the characteristic polynomial of the linear recurrence $x_{n+2}+2x_{n+1}+2x_n=0$ (or $x_{n+2}=-2x_{n+1}-2x_n$, that is $x_{n+2}=x_{n+1}+x_n$) and the sequences generated are $(0,0,...)$ (periodic with period $1$) and $(0,1,1,2,0,2,2,1,0,...)$ (periodic with period $8$). Any other sequence is a cyclic shift of these two.

My question is the following: let $K$ be a field that contains ${F_q}$ and in such a way that $f(t)$, as an element of $K[t]$, is also irreducible. There is a result (6.28 in Finite Fields, of Lidl & Niederreiter) that asserts that every non-null sequence generated by such an $f(t)$, with elements in $K$, is periodic with period equal to $ord(f(t))$ (in this case, $q^r-1$).

Note that these sequences can also be viewed as belonging to $K^r$.

But besides the lenght of these sequences being the same, anything more of interest can be said about these sequences? I know this is this rather vague. What I would like to prove (or disprove) is the following: in $K^r$ all sequences that have $f(t)$ as a characteristic polynomial are non-null (every element of it is non-zero) or alternatively is a multiple of (a cyclic shift) of the unit sequence? Probably I am asking too much: I am asking that, as soon as the sequence has a zero, it must be a multiple of the unit sequence.

Note also that this only concerns polynomials of degree $r \geq 3$ (if $r=2$ this is true).

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    What may be of interest to you is the fact that the sequences gotten using different primitive polynomials, in addition to sharing the period $L=q^r-1$, are *decimations* of each other. In other words, if a decimation exponent $d$ is coprime to $L$, and one primitive polynomial gives you the sequence $x_0,x_1,x_2,\ldots$, then we can construct another primitive polynomial giving you the sequence $x_0,x_d,x_{2d},\ldots$ (calculate multiples of $d$ modulo $L$). – 2017-02-24
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    Also observe that $f(t)$ being irreducible does not guarantee that $L$ is the smallest period. The simplest example is the polynomial $f(t)=t^4+t^3+t^2+t+1$ that is irreducible in $F_2[t]$ but not primitive, because $f(t)\mid t^5-1$. The sequences it generates will have period five. – 2017-02-24
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    Also look up $m$-sequences. Those have the so called *shift-and-add*-property: if you add two distinct shifts of an $m$-sequence, you get a third shift. A primitive polynomial gives you an $m$-sequence IIRC. Solomon Golomb's book on LFSRs is a classic, and explains these things in detail. – 2017-02-24
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    Thank you for your help. I was not aware of the shift-and-add-property; it might help... I will also try to find Golomb's book. – 2017-02-25

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