I'm reading this book on Galois Theory, and, while doing so, I've encountered a sentence that doesn't seem to make sense for me.
"Theorem: every polynomial $a$ in $K[t]$ is congruent modulo $m$ to a unique polynomial of degree less than the degree of $m$"
My issue with this:
Take $a$ such that it's degree is smaller than that of $m$. Then, if we assume that the author is correct, $a=pm+r$, where the degree of $r$ is also smaller than that of $m$. By the division algorithm, $p=0$, so we are left saying that $a-r = 0$ divides $pm = 0$.
So I wonder, is this statement the author made correct?
I would really appreciate any help :)
The author's statement is correct.
Two polynomials are congruent, mod $m$, if their difference is divisble by $m$.
If $a=r$, then $a$ is congruent to $r$, mod $m$, since $a-r = 0$, and $0$ is divisible by $m$.
So in your discussion, you have two errors.
Firstly, the test is for divisibility by $m$, not by $pm$.
Secondly, you have the order reversed.
The test is not to see if $a-r$ divides $m$. Rather, it's to see if $m$ divides $a-r$.