Compute the greatest common divisor of the ideals and in .

Recall that . Since , this ideal contains 1, and so .

unnecessary lemmas. very sloppy. handwriting needs improvement.

Compute the greatest common divisor of the ideals and in .

Recall that . Since , this ideal contains 1, and so .

Find the prime factorizations of the ideals and in . Compute .

Evidently and , so that and . Since is a unique factorization domain, , , , and are prime since 2, 3, 5, and 7 are prime.

Now , since , , and .

Let such that is maximal as an ideal in . What can be said about and ?

Since is a principal ideal domain, for some . If is maximal, then it is prime, so that is a prime element. Moreover, is a greatest common divisor of and . So is prime.

Compute the greatest common divisor of and in .

We will carry out the Euclidean algorithm. Note that , , and . In particular, is a greatest common divisor of and in . Since 22 is a unit, we see that is also a greatest common divisor.

Indeed, and . Note that is irreducible in (hence ) since has no integer solutiions, and similarly is irreducible, and these are not associates.

Let be a field and let be relatively prime. Prove that and can have no roots in common.

We noted previously that is a Bezout domain. In particular, if and are relatively prime then there exist such that . If is a root of both and , then we have , a contradiction.

Prove that if and are relatively prime in , then they are relatively prime in .

Suppose are relatively prime. Then there exist such that . Suppose now that is a common divisor of and ; say and . Then in , so that is a unit. In particular, and are relatively prime in .

Fix a natural number . A *residue system* mod is a set of integers such that every integer is congruent mod to a unique element of . Suppose and are both residue systems mod ; prove that and are relatively prime.

First suppose . In this case, the only residue system is . Certainly then is only a residue system if ; thus 1 is a greatest common divisor of and , and hence these two are relatively prime.

Now suppose and that and are residue systems mod . In particular, there exists such that mod . Say , so that . Thus 1 is a greatest common divisor of and , so that these are relatively prime.

FInd all of the nonunit, nonassociate, and nonconjugate common divisors of and in .

Note that . Since 3 is irreducible and does not divide , we have . Note that , and that these factors are irreducible since their norms are prime. Now , and these factors are also irreducible. Thus the greatest common divisor of and is . Since this element is irreducible, it is (up to associates) the only nontrivial common divisor of and .

Compute the greatest common divisor of 123 and 152. Moreover, find and such that .

By repeated application of the division algorithm, we see that , , and . Thus . Back-substituting to eliminate 29s and 7s, we see that .

Determine whether 7 and 0 are relatively prime. More generally, compute their greatest common divisor.

Note that 7 divides 7 since and 7 divides 0 since . Now suppose is an integer dividing both 7 and 0; certainly divides 7. Thus 7 is a greatest common divisor of 7 and 0.

In particular, since , 7 and 0 are not relatively prime.