Algebraic Formal Power Series

Question

I a quite confused by the wording of the following problem:

Consider the formal power series $R(X) = \sum_{n\geq 1} r_n X^n$ over $GF(2)[[X]]$. Show that $R$ is algebraic over $GF(2)[X]$ by deriving an algebraic equation, with polynomial coefficients, of which R is a root.

Here $GF(2)$ is the Galois field with two elements, and $r_n = 0$ if and only if n is of the form $4^a(8k+7)$, otherwise $r_n = 1$ (this sequence).

My abstract algebra may be a bit lacking for this course. Essentially what the problem is asking for is a algebraic equation

$$ A(X) = \sum_{n\geq0} a_n X^n = 0 $$

Where each $a_i \in GF(2)[X]$ is a polynomial, and $R(X)$ is a root of $A(X)$.

Am I interpreting the wording correctly? To be a little more explicit, the use of the word "over" is what confuses me the most. In this context what does it mean to be algebraic "over" something?

Answer 1

Your interpretation is more or less correct. In general, if $C$ is a commutative ring and $D$ is a subring of $C$, then an element $c\in C$ is said to be algebraic over $D$ if there exists a polynomial $f(T)\in D[T]$ (not the zero polynomial) such that $f(c)=0$. The "over $D$" just means that the coefficients of $f(T)$ must be elements of $D$. In your case, $C=GF(2)[[X]]$ and $D=GF(2)[X]$.

So in this case, that means you want a nonzero polynomial $f(T)$ whose coefficients are elements of $GF(2)[X]$ such that $f(R(X))=0$. This is almost what you wrote with your $A(X)$, but there are a few points to be careful of. First, $f(T)$ must be a polynomial, which means that your sum $\sum_{n\geq0} a_n X^n$ should be a finite sum (I'm not sure if this is what you had in mind). Second, $f(T)$ must be nonzero, so at least one of your $a_n$ must be nonzero (otherwise $R(X)$ would trivially be a root no matter what). Finally, I would strongly recommend using a variable different from $X$ (as I have done using $T$) to avoid confusion with the use of $X$ as an element of $GF(2)[[X]]$.