By Goodman F.M.
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This self-contained account of deformation conception in classical algebraic geometry (over an algebraically closed box) brings jointly for the 1st time a few effects formerly scattered within the literature, with proofs which are rather little identified, but of daily relevance to algebraic geometers.
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This completes our treatment of unique factorization of polynomials. Before we leave the topic, let us notice that you haven’t yet learned any general methods for recognizing irreducible polynomials, or for carrying out the factorization of a polynomial by irreducible polynomials. In the integers, you could, at least in principle, test whether a number n is prime, and find its prime factorsp if it is composite, by searching for divisors among the natural numbers Ä n. For an infinite field such as Q, we cannot 54 1.
That is, if Œa is invertible, can there be two distinct elements Œb and Œc such that ŒaŒb D Œ1 and ŒaŒc D Œ1? 7. If a nonzero element Œa of Zn is a zero divisor, can there be two distinct nonzero elements Œb and Œc such that ŒaŒb D Œ0 and ŒaŒc D Œ0? 8. Write out multiplication tables for Zn for n Ä 10. 9. Using your multiplication tables from the previous exercise, catalog the invertible elements and the zero divisors in Zn for n Ä 10. Is it true 44 1. ALGEBRAIC THEMES (for n Ä 10) that every nonzero element in Zn is either invertible or a zero divisor?
13. m; n/. Proof. 8. 14. The integers 21 and 16 are relatively prime and 1 D 3 21 C 4 16. 15. If p is a prime number and a is any nonzero integer, then either p divides a or p and a are relatively prime. Proof. 9. ■ From here, it is a only a short way to the proof of uniqueness of prime factorizations. 6. 16. Let p be a prime number, and a and b nonzero integers. If p divides ab, then p divides a or p divides b. Proof. 13. Multiplying by b gives b D ˛ab C ˇpb, which shows that b is divisible by p.
Algebra. Abstract and concrete by Goodman F.M.