Who's responsible for gettng rid of a rodent, the tenant or landlord?

As a general matter, it is the landlord's responsibility to insure that rented premises are inhabitable, which includes pest removal or extermination; even in the absence of a specific lease provision to that effect, this would be mandated by the "implied warranty of habitabilty" which is built into all residential leases. However, there is often an exception is the rodent (or insect, or mold, etc.) problem can be shown to be caused by the tenant. Therefore, if the problem is that you, as tenant, do not take out the garbage, leave food laying around, or otherwise cause the problem, in that case, the landlord might not be responsible for dealing with it. Also, the landlord might not be responsible if a tenant has refused to cooperate with prior extermination, etc. attempts. However, other than that, exterimation and other pest removal issues are the landlord's responsibility.