Yeah, if it's a decorative vase with no filter or media, carbon, etc...It won't ever begin to build good bacteria that eats ammonia. With no filter a tank will never cycle.
Are you feeding the betta meaty pellets?? It isn't one of those plant growing vases is it? BC they need meaty food or they will starve.
The tanks I have with NO FILTER I change 25% everyday or every other day so the ammonia doesn't go crazy, plus I use PRIME or KENT MARINE tap water conditioner.
If you have $25-30 bucks for a 5 gallon Hawkeye hex tank from Walmart.com, get it... then get a $1 algae sponge to baffle + rubber band for the filter which is too strong for betta.... throw a bit of cocktail shrimp (without the betta in it) for like a week or three (what it took my 16G to cycle)...it doesn't smell or anything either which I thought it would....this or the eclipse 3 galllon is one sale on amazon for $40, and it stays 79-81 without a heater.
Hope this helps!!! Good luck!!