Make sure your filter has some type of biomedia in it. Like 30ppi foam from Amazon.
You will have ammonia issues unless your filter has a foam with enough surface area to grow the bacteria that you need.
If your filter already has a foam in it, then you're fine. Just use Prime daily whenever Ammonia is at 0.25ppm and water change if you ever see 0.5ppm. (Until you get Prime, you will have to water change immediately at any ppm Ammonia). It takes a while to cycle a tank, potentially 1-3 weeks.
Also, Snails can't catch Ich. So you're good there. It's possible the Ammonia was just from the dead Tetra and since they're gone you won't have an issue anymore.
Good luck!