Chapter 1, "Why Fields are Prime"

[chanhosuh]

The general note section called "Why Fields are Prime" is problematic.

1. The title, "Why Fields are Prime", is incorrect, as a field's order may be a prime power with power greater than 1. This is acknowledged several times in the text, including in the first sentence of the section.

2. The first sentence of the section itself is a bit strange. It seems to suggests the phenomenon appearing in Exercise 5 has some bearing on the order of a finite field being a prime power. I can't make heads or tails of this.

3. I suspect the exposition of the chapter may need to be changed somewhat to reflect that the ring given by modular arithmetic isn't a field unless the modulus is prime. For example, what you might consider "F_9" (really Z_9) doesn't have nice properties like in Exercise 5 but only if you insist upon trying to use modular arithmetic (of course, under the true field multiplication, F_9 does behave nicely under multiplication by a non-zero element).

Suggested corrections/updates:

• Change title to: "Why Prime Fields are Useful"
• Change first sentence to: "The answer to Exercise 5 is why we choose to use finite fields with a prime number of elements."
• I think this section would make more sense if there were a short discussion about when Z_n (ring given by mod arithmetic) is a field (only happens when n is a prime)

P.S. Great book, really enjoying it so far!

[jimmysong]

Thank you! I'm not sure if I can include a whole section for Z_n being a field only when n is prime, but that's something I'll consider for the second edition.

The first two suggestions, I'll use. Thanks!