Neukirch.tex (worked in June 2021)

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Book: Algebraic Number Theory, J. Neukirch Document: Books/Neukirch/Neukirch.tex Date: June 2021

Exercise 1: okay, minor improvement: alpha \overline{alpha} is non-negative real.

Exercise 2: 呢節未講到Z[i]係UFD, 所以個proof開首唔係咁好……後面文字亦寫得唔清楚。

Exercise 3: 大方向正確，但Z[i]裏唔可以用gcd, 要有order先可以講greatest ，但Z[i]唔係ordered ring。 應該let r in Z[i] and r divides both x+yi and x-yi, 然後嘗試去證 r 屬於 units 。 用輾轉相除法去證 ( x+yi, x-yi) ~> (x+yi, 2x) ~> (x+yi, x) ~> (y, x) [後面兩節我加嘅, 注意呢道寫法亦唔formal]　

anyway, 未prove完？有 x + yi = e alpha^2, 之後再諗到埋 x - yi = e' beta^2, 然後solve二元一次方程式系統，咁就差唔多完成。

之後嘅" x + yi &=& e \alpha^2 \ &=& e (u + vi)^2 \ &=& e (u^2 + 2uvi -v^2) " , very good .

Exercise 4: 唔應該用 ordered field axioms。 Z[i]係ring，唔係field。

PS 咦我發現我個pdftex rendering有啲問題…

Notes:

Algebraic Number Theory 我接觸過呢本開頭，覺得幾好：

《Introductory Algebraic Number Theory》 / Şaban Alaca and Kenneth S. Williams (2003)

https://www.cambridge.org/core/books/introductory-algebraic-number-theory/9F53B233CD4D717B1A31ECD117FFEA7D

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原來應​該用 pdflatex ...

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Exercise 5 okay

Exercise 6 okay

Exercise 7 Prime elements in Z[sqrt 2]漏咗一個case: sqrt(2) :)

最尾個句有typo ... are either integer primes with 2 not a quadratic residue mod p or solutions to the generalized Pell’s equation x^2-2y^2 = p with p 2 a quadratic mod p.

( My Reference: Prove that Z[sqrt 2]is a Euclidean domain https://math.stackexchange.com/questions/150885/proving-that-mathbbz-sqrt2-is-a-euclidean-domain

Prime elements in Z[sqrt 2] https://math.stackexchange.com/questions/674814/prime-elements-in-mathbbz-sqrt2 )