Abstract
We have already used N(xy) = N(x).N(y) in the previous chapter, for the “norm” N(a + bi) = a 2 + b 2 in ℤ[i]. The units in ℤ[i] being {±1, ±I} i.e. the elements x ∈ ℤ[i] with N(x) = 1 we infer that x is prime in ℤ[i] iff N(x) is prime in ℤ (indeed, the no-associated decompositions correspond to each other). Hence N(1 + i) = 2 implies that the ideal (1 + i) is prime.
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© 1998 Springer Science+Business Media Dordrecht
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Cǎlugǎreanu, G., Hamburg, P. (1998). Prime Ideals, Local Rings. In: Exercises in Basic Ring Theory. Kluwer Texts in the Mathematical Sciences, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9004-4_30
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DOI: https://doi.org/10.1007/978-94-015-9004-4_30
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4985-8
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