Ttgjyuk

There are two propositions in the chapter of Number Theory in my book, the proofs of which I am having trouble to understand.

For Proposition 3

I cannot understand the proof from "Therefore ..." in the third line. I was thinking that maybe "$m-l$ divides $k$" will be "$m-l$ divides $n$". Also, is $k/d$ an integer?
Also, can someone please give an example to clarify this theorem?

For Proposition 4

What does $f(x) in Z[x]$ mean? Why are the third brackets used?

I cannot understand how "By proposition 2, $a_ja^j equiv b_jb^j$ (mod $n$)...".
Can someone please explain?

For proposition 3, I agree with fleablood's assessment in a comment above that this is probably a typo.

For proposition $4$, note that the thing you're asking about appears in a parenthesis, following the abbreviation "i.e.". That means that even without knowing what "$fin Bbb Z[x]$" means, we can infer that it's merely an alternative formulation of whatever came before, namely that $f$ is a polynomial with integral coefficients.

To actually answer the question, given a ring $R$, the notation $R[x]$ means "the ring (or set) of polynomials with coefficients in $R$". So $Bbb Z[x]$ is the ring of polynomials with integer coefficients. And $fin Bbb Z[x]$ means exactly that $f$ is an element of this ring.

As for why we use the brackets there? That's just convention. You could have $Bbb Z(x)$ as well, but that usually means the ring (or set) of rational functions with integer coefficients.

Finally, the "by proposition 2" thing, note that proposition 2 states that if we have a product of two things, and we change one of the factors to a congruent factor, the product is unchanged modulo $n$. So $$a_ja^j = a_ja^j-1cdot aequiv a_ja^j-1bpmod n$$
So we can swap one $a$ for a $b$. Now just swap the other $j-1$ $a$'s for $b$'s, one by one as proposition 2 says you're allowed to do, and finally you reach $a_jb^j$.

I think the book made a typo and it isn't $m-l$ that divides $k$ (I don't see that as even true) but that $d$ divides $k$.

This follow as $d$ divides $kn$ but is relatively prime to $n$ so $d$ must divide $k$.

And hence, yes, $frac kd$ is an integer, for which we conclude $m-l equiv 0 pmod n$.

Note: This will not be true if $a, b, n$ will have a common divisor (other than $1$). Consider $8 equiv 20 mod 12$ but $2equiv 5 mod 12$ is .... wrong. (Although $2 equiv 5 pmod 3$.....)

--- I empathize. For a typo that is a doozy to make and utter destroys the intent of the proof.

....

$mathbb Z[x]$ means the set of all polynomials with integer coefficients.

So $f(x) in mathbb Z[x]$ means "Let $f(x)$ be a polynomial with integer coefficients.