Can We Actually Terraform Mars?

TL;DR Summary
Transforming the Red Planet into a second Earth is a staple of science fiction. But is it scientifically possible?
The Challenge
Mars is currently a frozen, irradiated desert. Its atmosphere is less than 1% as thick as Earths, and it lacks a global magnetic field to protect against solar radiation. To terraform Mars, we would need to thicken the atmosphere, raise the temperature, and introduce a biosphere.
Warming the Planet
The first step is to release greenhouse gases. Some scientists have proposed detonating nuclear weapons over the Martian poles to vaporize the dry ice (frozen CO2) trapped there. This could trigger a runaway greenhouse effect, warming the planet and melting subsurface water ice.
The Magnetic Problem
Even if we could create a breathable atmosphere, the solar wind would gradually strip it away because Mars has no magnetic field. Proposed solutions include building a massive magnetic shield at the Mars L1 Lagrange point to deflect the solar wind, allowing the atmosphere to regenerate naturally.