playing field vs arena

playing field and arena both are nouns.

Word	Noun	Adjective	Verb	Adverb
playing field	Yes	No	No	No
arena	Yes	No	No	No

As nouns, arena is a hyponym of playing field; that is, arena is a word with a more specific, narrower meaning than playing field:

playing field (noun)	arena (noun)
a piece of land prepared for playing a game	a playing field where sports events take place
the circumstances under which competition occurs	a large structure for open-air sports or entertainments
	the central area of an ancient Roman amphitheater where contests and spectacles were held; especially an area that was strewn with sand
	a particular environment or walk of life