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I am reading a text on finite fields and got stuck on a little side note of the introductory part. I hope I am adding enough context, if not let me know. $p$ is a prime number and $a,b \in \mathbb Z$.

For every integer $a$ we denote with $\bar a\in {0, ..., p-1}$ the remainder of $a$ when divided by $p$. That means $\bar a$ is a uniquely determined number in ${0, ..., p-1}$ with $a=kp+\bar a$ for a $k\in \mathbb Z$. (This has already been proven) ... It is easy to prove that $\overline{a+b}=\bar a +\bar b$ and $\overline{ab}=\bar a \bar b$.

Unfortunately this is not easy for me. My try:

Let $a=kp+\bar a$ and $b=lp+\bar b$ with $k,l \in \mathbb Z$. $$a+b=kp+\bar a + lp+\bar b;$$ $$a+b=(k+l)p+(\bar a+\bar b).$$

Thus $\overline{a+b}=\bar a +\bar b$. However, it may well be that $\bar a +\bar b$ larger than $p$, so this does not work.

While looking for an answer I found this question: Prove that $\overline{a+b}=\bar{a}+\bar{b}$ But this is not helping me.

Carlevaro99
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  • Hello :) Your $\bar{a}$ does not need to be less than $p$. It only says there is a representative $a'$ with $0\leq a'<p$. Maybe one has to distinguish between the coset and the representative. Commonly, you set $\bar{a}:={a+kp\mid k\in \mathbb Z}$ and then your equation holds. – Jochen Jul 23 '21 at 11:32

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You are right and the book is wrong.

The best you can get with the definition of $\bar a$ you quote is $$ \overline{a+b} = \overline{\bar a + \bar b} \qquad\qquad \overline{ab} = \overline{\bar a \bar b} $$ or alternatively $$ \overline{a+b} = \bar a \boxplus \bar b \qquad \qquad \overline{ab} = \bar a \boxdot \bar b$$ where $\boxplus$ and $\boxdot$ are modular addition and multiplication operations on $\{0,1,\ldots,p-1\}$.

The modular operations are more commonly defined to work on residue classes instead of on the particular representatives $\{0,1,\ldots,p-1\}$. It feels plausible that the author of your text was confusing what they were doing with a notation where $\bar a$ means $\{np+a\mid n\in\mathbb Z\}$, the residue class containing $a$. But even so, that requires an explicit definition of how to add and multiply such classes.

Troposphere
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