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The work referred to is Ireland and Rosen A classical introduction to modern number theory, GTM 84, 2nd ed.

On page 1 prime is defined as follows: a number $p$ is a prime if its only divisors are 1 and $p$. That makes 1 a prime which is surely not intended. On page 2, where primes in $\mathbb Z$ are discussed, I&R do exclude $\pm1$. On page 3 they claim Every nonzero integer can be written as a product of primes. I am prepared to believe that 1 is a product of primes, i.e., it is the product of the empty set (of primes), but I think $-1$ has to be excluded. I don't think I&R are simply being careless, so what am I missing?

Justin
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    This is indeed sloppy, as they should have said $p\geq 2$. They somewhat imply this in the next paragraph with "The first prime numbers are 2,3,5,..." – Alex R. Aug 25 '20 at 18:20
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    I think they are being careless, and I also think they are writing for a readership who already know very well what prime numbers are. – Angina Seng Aug 25 '20 at 18:21
  • Leading up to that passage, they say, "If we are given a number, it is tempting to factor it again and again until further factorization is impossible. [...] Numbers that cannot be factored further are called primes." Okay, so $2 = 1 \times 2 = 1 \times 1 \times 2 = \ldots$, therefore $2$ is not a prime. They're simply being careless. – Théophile Aug 25 '20 at 18:23
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    This imprecision has been noted here before at length, e.g.see this answer.. If your question is answered there then let me know and I'll close this as a dupe. If not please clarify what remains unanswered. – Bill Dubuque Aug 25 '20 at 18:49
  • @Gone - my question is certainly related to what you cite, and there is a good discussion there! I have this worry: sloppiness on the first three pages I can recognize and fix for myself, but what if the sloppiness continues throughout the book? But that's not a question for stack exchange, it's just a warning to myself. Do close my q. if you wish to. – Justin Aug 25 '20 at 20:10

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