By David M. Bressoud
"About binomial theorems I'm teeming with loads of information, With many pleased proof concerning the sq. at the hypotenuse. " - William S. Gilbert (The Pirates of Penzance, Act I) The query of divisibility is arguably the oldest challenge in arithmetic. old peoples saw the cycles of nature: the day, the lunar month, and the 12 months, and assumed that every divided lightly into the following. Civilizations as separate because the Egyptians of 10000 years in the past and the significant American Mayans followed a month of thirty days and a 12 months of one year. even if the inaccuracy of a 360-day yr grew to become obvious, they most well liked to continue it and upload 5 intercalary days. The quantity 360 keeps its mental attraction this present day since it is divisible through many small integers. The technical time period for one of these quantity displays this allure. it's known as a "smooth" quantity. on the different severe are these integers without smaller divisors except 1, integers that can be known as the indivisibles. The mystic characteristics of numbers comparable to 7 and thirteen derive in no small half from the truth that they're indivisibles. the traditional Greeks discovered that each integer may be written uniquely as a fabricated from indivisibles higher than 1, what we accurately name top numbers. to understand the decomposition of an integer right into a made from primes is to have an entire description of all of its divisors.
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N and α1 , . . , αn of OK . Show that ∆(α1 , . . , αn ) = ∆(α1 , . . , αn ). 22. Let K = Q( d) be a quadratic number field with d a squarefree integer. Show that d d ≡ 1 (mod 4); ∆K = 4d d ≡ 2, 3 (mod 4). CHAPTER 3 Prime Splitting In this chapter we will investigate how to explicitly factor ideals in rings of integers of number fields. A common theme will be considering ideals of one ring in another. Specifically, we will often have the following situation: K and L will be number fields with K ⊆ L (so OK ⊆ OL ), a will be an ideal of OK , and we will consider the ideal aOL of OL generated by a.
Since a21 − b2 d ∈ Z 4 we must have b = b1 /2 where b1 in Z is also odd. Substituting this in, we find that a21 − b21 d ≡ 0 (mod 4), this being an ordinary congruence over the integers. Now, since a1 and b1 are both odd, a21 ≡ b21 ≡ 1 (mod 4). Substituting these in, we find that 1 − d ≡ a21 − b21 d ≡ 0 (mod 4), 2. ALGEBRAIC INTEGERS 41 so d≡1 (mod 4). Thus in the case that d ≡ 2, 3 (mod 4) there are no algebraic integers with a half an odd integer; if d ≡ 1 (mod 4), then there are additional integers of the form √ a1 + b1 d 2 where a1 and b1 are odd.
We have 5 = (2 + i)(2 − i) and 13 = (3 + 2i)(3 − 2i). 1. UNIQUE FACTORIZATION 31 To finish the factorization we simply have to figure out which of the prime factors of 13 divides α and whether one or both of the prime factors of 5 divide α. One finds that (2 + i)2 and 3 + 2i divide α. Up to a unit, then, the factorization of α is (1 + i) · (2 + i)2 · 7 · (3 + 2i); multiplying it out we find that the unit is i, so that −133 − 119i = i · (1 + i) · (2 + i)2 · 7 · (3 + 2i). Our analysis of factorization in Z[i] seems to have suggested some connection with primes of the form x2 + y 2 .
Factorization and Primality Testing (Undergraduate Texts in Mathematics) by David M. Bressoud