I think that if we could build a powerful enough laser/maser to heat the core of Mars up and melt it, it could generate a magnetic field, which would allow for easier living on Mars. Could this be done?
This question assumes Mars doesn't have a magnetic field because Mars' core is frozen solid. It's not. Mars has a partially liquid core, just as does the Earth. (The Earth has a liquid outer core and a solid inner core.) Whether Mars has a solid inner core is unknown, but it certainly does have a liquid outer core (and possibly a fully liquid core). (See C.F. Yoder, et al. "Fluid core size of Mars from detection of the solar tide." Science 300.5617 (2003): 299-303.)
Three (or more) ingredients are needed for a planet to have a substantial magnetic field:
A conductive fluid. Mars has that. It has, at a minimum, a molten outer core.
Planetary rotation. Mas has that as well.
A significant heat flux across the core-mantle boundary. Mars does not have that. (Nor does Venus.)
Mars has a stagnant lid geography. (So does Venus.) This keeps the heat bottled up in the core. A planet with a stagnant lid doesn't have enough heat flux out of the core to maintain a magnetic field.
If you want Mars to have a magnetic field, you'll have to unloose the heat that is bottled up in Mars' core. That means either rampant vulcanism (the best way a terrestrial planet can release heat into the universe) or plate tectonics (the second best approach).
XKCD has an interesting discussion of a similar question, aiming lasers at the Moon instead of Mars.
- it takes 5 PW (double of Earth's current electric power production) to light up the moon with the same brightness as the full moon.
- it takes 50 EW to heat up the surface noticeably.
Mars is bigger than the Moon, but we are within a few orders of magnitude of what'd be required. So we'd need 10,000 times the total power output of all electric power stations on Earth to heat up Mars. The laser itself would also have to be gigantic, you'd end up building a planet-sized ship to house the laser (yep, a Death Star).
And as PearsonArtPhoto said, you can't heat the core without heating the surface. As David Hammen explains, that's not inconvenient, you want to melt part of the surface anyway to create plate tectonics.
When you're done, it'd take thousands of years for the surface to cool enough to be livable, so this would be a really expensive and inefficient method of geoengineering. You'd be better off just building the Death Star without the laser and living on that.
Not without also heating the surface. And such a beam would have to be REALLY powerful.
All electromagnetic waves are either absorbed or reflected by Mars. Thus, any heat that was applied via the beam would also heat the surface. The only heating of the core would come via indirect methods.
I don't think anything short of somehow drilling to Mars's core would allow for us to heat it any.