Other forms: intransigently
Intransigent means inflexible, stubborn, entrenched. Argue all you like with an intransigent three-year-old. He will never back down from the position that he wants the lollipop NOW.
Trans has to do with movement — think transportation, or a package in transit, i.e. "on the way." The in- of intransigent means "not," so something or someone who is intransigent is not moving. If one political party wants to raise funds to improve schools but the other is intransigent on the subject of higher taxes, the debate will get nowhere.