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Factoring in the Gaussian Integers- Further Questions (Algebra 2: Lecture 7 Video 4)
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Lecture 7: In this lecture we focused on factoring in the Gaussian integers Z[i]. Since Z[i] is a Euclidean domain, it is a PID and a UFD. We also know that an element in Z[i] is irreducible if and only if it is prime. We started by talking more generally about quadratic integer rings and their associated norms. We saw that if the norm of an element is a prime or the negative of a prime, that element is irreducible. We then considered a prime element pi and saw that (pi) intersect Z equals pZ for some prime integer p. We saw that in order to find all prime elements in this quadratic integer ring, it is enough to consider the irreducible factors of the integer primes p in this ring. We saw that there were two possibilities for how an integer prime factors-- either it remains irreducible, or it factors as a product of two irreducible elements, both of which have norm p or negative p. For the rest of the lecture we focused on the special case of Z[i]. We saw that a prime p factors into a product of two irreducible elements if and only if p is the sum of two integer squares. We saw that primes congruent to 3 modulo 4 cannot be written as a sum of two squares. We stated Fermat's theorem on the sum of two squares and a result giving a complete characterization of the irreducible elements of Z[i] (up to associates). We stated and proved a corollary characterizing not only which integers n can be written as a sum of two squares, but in how many ways an integer can be written as a sum of two squares. We then proved that primes congruent to 1 modulo 4 can be written as a sum of two squares. At the end of the lecture we discussed how the questions from this lecture generalize and lead to some interesting questions in number theory.
Reading: The main reference for this lecture is Section 8.3 pages 289-292. We did things a little out of order by first assuming that a prime p that is congruent to 1 modulo 4 can be written as a sum of two squares and using this fact to prove Proposition 18. We then came back and proved this statement at the end, following the proof of Lemma 17 given on pages 290-291.
We stated some results that are coming up soon that will make this argument a little easier, Proposition 17 and Corollary 19 in Section 9.5. In the last video we said a little about how these questions generalize to other rings and mentioned some of the results of Section 16.3. You do not need to know anything about Dedekind domains for the rest of this course, but if you are interested in these kinds of number theory problems, I recommend that you take a look at this section.
Reading: The main reference for this lecture is Section 8.3 pages 289-292. We did things a little out of order by first assuming that a prime p that is congruent to 1 modulo 4 can be written as a sum of two squares and using this fact to prove Proposition 18. We then came back and proved this statement at the end, following the proof of Lemma 17 given on pages 290-291.
We stated some results that are coming up soon that will make this argument a little easier, Proposition 17 and Corollary 19 in Section 9.5. In the last video we said a little about how these questions generalize to other rings and mentioned some of the results of Section 16.3. You do not need to know anything about Dedekind domains for the rest of this course, but if you are interested in these kinds of number theory problems, I recommend that you take a look at this section.