I've been doing many problems that go like here's a field verify this is a subfield or here's a domain, verify this is a subdomain. However its often very tricky for me to see it and write clear proofs. Is there is a list of criteria you can recommend that I can use to prove subfield and subdomain.
For instance, to check for subfield S given field F we need to show 0 is in S, ab is in S and a-b is in S.
Are there criteria like these that I can use to check for subfield or subdomain?