you need to be clearer about what Q[[t]] is. usually, this notation means all formal power series, not polynomials (a polynomial is a "finite power series").
it is true that all units in Q[t] (the ring of rational polynomials) are elements of Q*.
but let's consider what we have in power series.
consider the power series:
1 + t + t2 + t3+...+ tn +.....
what happens when we multiply this by 1 - t?
(1 - t)(1 + t + t2 + t3+...+ tn +.....)
= (1 + t + t2 + t3+...+ tn +.....) - (t + t2 + t3+...+ tn +.....) = 1 + 0t + 0t2 + 0t3 +.... = 1.
this is not a statement of convergence, we are not considering "values" for t, it is a statement about FORMAL power series.
the general problem of determining the units in Q[[t]] is a bit more complicated. if
is a unit of Q[[t]], there is some element:
with ur = 1.
we have, for the product:
if this is to be equal to 1, then c0 = 1, ck = 0, for k > 0.
now c0 = a0b0, so we see a0 cannot be 0, if u is to be a unit. if a0 ≠ 0, define the bk like so:
verify that this definition makes r an inverse for u.