Hi there. I know what properties are necessary for a ring and for a field. I'm very poor at formal proofs however. Could someone help me learn how to formally prove whether a set is a ring or a field? Here are some questions, pick whichever one you would like to answer

1. Show that the following sets are rings. Determine whether they are commutative and if there is a unity.

- Z[x], the set of all polynomials in x with integer coefficients.
- The set of all 3x3 matrices with real entries.
- The set of all even integers n ∈ Z.

2. Show that the set Q(√2) = { a + b√2 | a, b are rationals } forms a field.

First I'll try my best to answer one of them:

1C)

let n = 2m or 2k where m and k are integers

2m + 2k = 2(m+k) since 2(m+k) is even we have additive closure

2m * 2k = 4mk = 2(2mk) since 2(2mk) is even we have multiplicative closure

*I'm not exactly sure how to prove the other properties, or even if I have to? Can't I just assume this set has them because integers have them, and this is a set of integers? It seems obvious after determining closure that this set is both a ring and a field