We talk about Gauss primes and find a complete answer to the question when a prime integer will remain a prime in the ring Z[i]. In fact, we prove that, for a prime integer p the following are quivalent:
1. p is the product of a Gauss prime and its complex conjugate
2. p is equal to the sum of two squares of integers
3. The polynomial x^+1 has a root modulo p
4. p= 2 or congruent to 1 modulo 4.
In view of the above, a prime integer p is also Gauss prime if and only if it is congruent to 3 modulo 4.
After that, we talk about integral elements in an extension of commutative rings. We in fact prove three other equivalent conditions for an element to be integral in an extension of commutative rings.
