Let K be a field. Then K[[x]] denotes the ring of formal power series over K. I.e., {Sum_0<=j<oo} c_j x^j} for all sequences c = (c_0, c_1, c_2, c_3, …) of elements of K. Two elements of K[[x]] are defined as equal if an only if their coefficients of each power x^j of x are equal, j = 0,1,2,3,…. Addition and multiplication make sense (since only finitely many terms need be added to get the coefficient of x^j of the sum or product). So K[[x]] is a ring. It is easy to check that K[[x]] has no zero-divisors, so it is an integral domain, so has a field of quotients. I don't know the standard notation for this, but we can write Quo((x)) = {f(x)/g(x) | f, g in K[[x]]}, where we define f(x)/g(x) = h(x)/k(x) precisely when f(x) k(x) = g(x) h(x), as you might expect. * * * Interesting fact (exercise): The field Quo((x)) is identical to the formal Laurent series over K, namely Quo((x)) = {Sum_L<=j<oo} c_j x^j} for all sequences c = (c_L, c_(L+1), c_(L+2), c_(L+3), …) of elements of K where L is any integer (possibly negative). ------------------------------------------- QUESTIONS: --------- 1) In the simplest case — where the base field K = {0,1} — what is Quo((x)) ??? In other words, can we decompose Quo((x)) = quotients of formal power series in x over {0,1} into some kind of nice structure? It would in this vein be nice to know the prime elements of K[[x]] while we're at it. 2) And what can we say about the algebraic closure of Quo((x)) ??? —Dan