Buzz on microphone input when on AC, low volume

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I have a Toshiba L35-S2171, and I'm trying to record with a
microphone.  The volume isn't really high enough, even with every
volume control I can find set at max, and mic boost turned on.

But the big problem is that I get a loud buzz on this input when the
computer is plugged into the power mains.  When I go to battery
only, the buzz goes away.

The mic I'm using works ok on my desktop, but even there I have
trouble getting enough volume.  There is no buzz there.

From past experience, it seems to me that the buzz on AC is a sign
of poor design of the audio input circuit, or that the AC adapter is
really trashy.  I don't know that I can do much about either of
those.  However, when I plug a cable into the mic jack that's just a
male-to-male cable, no mic, then even on AC I don't get any buzz
unless I touch the plug at the other end.  This would suggest a
shielding problem with the mic itself, but that wouldn't explain why
there's no buzz using it on the desktop.

If anyone has a suggestion for a fix for the buzz on AC, I would
like to hear it.

I think it would help a lot if I didn't have to turn on mic boost.  
That would mean finding a mic that produces a much higher level than
the one I have, so that it would work at much lower volume settings.
Any suggestions for a particular mic like that?  (This is a very low
budget situation).)

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