I've been investigating the p-adic field, and I was wondering how the p-adics form a field.
The part I'm stuck at is that from what I've learned, to qualify as a field, every element except the additive identity, should have a multiplicative inverse. But, from what I've read, in the p-adic field, any p-adic integer that ends in zero doesn't have a multiplicative inverse. So, I snooped around abit and found another definition for a field, which said that every non-zero element has a multiplicative inverse. I'm assuming that this means that the p-adic numbers that end in zero would be considered as zero elements.
What exactly are zero elements? (Or non-zero elements?) and how does it apply to the p-adic numbers?