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Resistors up to a few Gohms can be bought. They introdude very little noise current so they can be used to bias the gate. The resistor in the drain circuit will have less effect on the noise because the signal is already amplified at that point. The resistor in the source, should be bypassed with a very large capacitor so that it drops out of the picture well below the 37Hz.
Try this:
Take one coin call heads +1 and tails -1. Flip it a bunch of times and take the RMS. (Or not bother because we know it comes out as one)
Take two coins and do the same adding the values.
On the average 1 time in 4 you get -2 and one time in 4 you get +2. The other 2 times you get zero. Therefor the RMS will come out to:
sqrt ( (2^2 + 2^2 + 0^2 + 0^2)/4 )
sqrt(2) = 1.414..etc
Any time you add random stuff together you get this sort of squarerooty thing happening.
The signal, however adds linearly so your signal to noise is improved by N/sqrt(N) or simply sqrt(N).
Ideal inductors are noise free. The resistances of an inductor make noise just like resistors. Magnetic cores make a thing called Barkhausen(sp) noise that sounds like snapping and poping on headphones. Inductors also tend to pick up AC magnetic fields and are also magnetic.
Your frequency is kind of low but take a look at Triad's or Tamura's small audio transformers.
Think more along the lines of getting up to 10 to 100 times not hundreds. Hundreds will not be easy to do and 10 or so should be enough. The amplifier does not need to be differential BTW.
You can make you pick up a tuned circuit to get some signal increase and filtering right at the start.