Question

the arms race meant that once the united states built hydrogen bombs,
the soviet union built them too.
world peace had been assured.
no other country wanted to build them.
a new world war was about to start.

Explanation:

The arms race was a competition between the US and the Soviet Union during the Cold War. In an arms race, if one side develops a new weapon (like hydrogen bombs), the other side typically feels compelled to do the same to maintain a balance of power. World peace was not assured as both sides having such powerful weapons increased tensions. It's not true that no other country wanted to build them (other countries did pursue nuclear capabilities later in different contexts, but in the immediate US - Soviet arms race context, the main competitor was the other superpower). And while tensions were high, it doesn't mean a new world war was immediately about to start just because the US built hydrogen bombs. The key concept is the reciprocal nature of the arms race between the two superpowers.

Answer:

the Soviet Union built them too.