Modular Arithmetic 2 - Inverse Element Clarification

[Tressos-Aristomenis]

As a first issue, let me know if any other format is more convenient.

[11/06/2023] Mentioned in Discord by @ NicoD.

Hello ! For the people who write the text for Modular Arithmetic 2. I think there could be confusion when talking about inverse element for addition and multiplication. We could understand from the text that the inverse is the same for addition and multiplication but it's not generally and also 0 as no inverse for multiplication

[hyperreality]

@GiacomoPope what do you think about this?

[GiacomoPope]

Yeah this could be reworded to be more precise. I'll do this now

[GiacomoPope]

Old text:

  A finite field <code>F<sub>p</sub></code> is the set of integers <code>{0,1,...,p-1}</code>, and under both addition and multiplication there is an inverse element <code>b</code> for every element <code>a</code> in the set, such that <code>a + b = 0</code> and <code>a * b = 1</code>.

New test:

  A finite field <code>F<sub>p</sub></code> is the set of integers <code>{0,1,...,p-1}</code>, and under both addition and multiplication there are inverse elements <code>b<sub>+</sub></code> and <code>b<sub>*</sub></code> for every element <code>a</code> in the set, such that <code>a + b<sub>+</sub> = 0</code> and <code>a * b<sub>*</sub> = 1</code>.

[GiacomoPope]

Note that we already had the comment:

Note that the identity element for addition and multiplication is different! This is because the identity when acted with the operator should do nothing: <code>a + 0 = a</code> and <code>a * 1 = a</code>.