Does the US have Nukes stationed in South Korea?

It is widely known that there is a huge minefield in the DMZ. As well as that there have been admissions by the US in the past that Nuclear landmines reside there as well( SADM's configured for land mine use). The question I have is are there actual nukes under US control in S Korea, should the North invade?

By nukes I mean artillery based nukes, rockets( short range), air delivered old school bomb type weapons and cruise missiles with nuke payloads.