I put together my first batch of beer, and it has been fermenting for about 11 days. For the most part I think everything is going quite well although I have been keeping the beer in a room that is less than optimal (temperature is 55 degrees or so). Taking all of this into consideration what is the minimum amount of time for fermenting beer?
A minimum time to ferment is a difficult question to answer as there is no absolute cut off point as regards fermentation. For example, cask or bottle conditioning of a brew is a continuation of fermentation over many weeks or months - and often by things other than sacchromyces yeast. A "better" question is probably "when do I know that primary fermentation has finished" although acknowledging the above comment it can be said that it never finishes. I suspect that what "finished fermenting" really means is that if the brew were bottled at that point, the bottles would not become "bottle bombs".
However the practical answer to "how do I know it has finished fermenting" is to measure the specific gravity of the brew over a few consecutive days. When the reading is static over two or even three days then the primary fermentation has "finished".
I have brewed using a hydrometer but over the years I have come to the conclusion that at "room temperature" (eg 16C-22C) a brew will ferment out in 14 days. I often pour the beer off the lees/trub at 10 days, leave it again for 4 days or so then mix in priming sugar and bottle. At lower temperature (eg 8C-14C) I will often leave the brew for 3 weeks in the primary fermentation vessel before pouring off the lees/trub and leaving to settle for another week before bottling. A lager can be left in the bin for 3 months before bottling and is the better for it.
The conditioning (anaerobic fermentation in the bottle) of the brew should be considered. A bottle of beer is, IMHO, not ready before one month in the bottle. And most improve over 6 months. Only very aromatic hoppy beers should be drunk earlier rather than later and that is only because the hop aroma and flavour mellows a lot over time.