Let p be a nonzero integer. Prove the set Z[sqrt(p)] = {b+[(sqrt(p)] | b,c in Z(integers)} is a subring of the complex numbers.
Ah! The Quadradic integers.
If p>0 then we have Z[sqrt(p)]={b+c*sqrt(b)}
We need to show that this structure forms a abelian group under addition.
Closure
(b+c*sqrt(p))+(b'+c'sqrt(p))=(b+b')+(c+c')sqrt(p)
Where (b+b') and (c+c') are integers.
Identity
b=0 and c=0.
Inverse
b=-b' and c=-c'
Associativitiy
Since C (complex numbers) are associate so are these.
Commutativity
Since C are abelian so are these.
Hence Z[sqrt(p)] form an abelian group under +.
We now show that Z[sqrt(p)] is closed under *.
Simple,
(a+bsqrt(b))*(a'+b'sqrt(p))=(aa'+bb'p)+(ab'+a'b)sq rt(p)
Where,
aa'+bb'p and ab+a'b are integers.
The last thing is two show that is is distributive.
Since C is distributive then so it Z[sqrt(p)]
When p<0 then actually it is undefined to take a squere root but anyway we define Z[sqrt(b)]=Z[a+ibsqrt(b)]
And use the same appraoch as above.