I will post here any corrections to the online notes which are found after
the start of term. Let me know if you spot anything wrong. I will give due
attribution unless people wish to remain anonymous!
I will not bother to enter here trivial errors which do not affect
the mathematical meaning and would not cause any confusion.
Page numbers refer to the full notes with proofs, not to the large format
notes without proofs used in lectures.
Introduction
Chapter 1
Section 1.5: On page 10, in the paragraph after Theorem 1.5.4, it says "and the
Euclidean algorithm may be used to compute them", but it is only in
a Euclidean Domain (ED) that we have such an algorithm, not in every
Principal Ideal Domain (PID). This is correctly stated soon after,
right after Proposition 1.5.7 (before the worked example).
Section 1.5, page 11, end of proof of Lemma 1.5.9: should say
$b=p(ub)+(ab)v$ (i.e. plus sign not minus).
Section 1.5, page 13, last example: the equation for $w$ should be
$w^2+w+5=0$, i.e. change $4$ to $5$.
Chapter 2
Section 2.4, proof of Prop. 2.4.3, line 3: $x^2\equiv-1\pmod{p}$,
not $\pmod 4$.
Section 2.5 (Chinese Remainder Theorem). The example in the
paragraph "Applications of CRT" on page 21 confuses some people,
possibly because I changed notation between page 19 and here. In
the sentence starting "To solve the second and third..." replace the
text in the notes by
write $1=7u+13v=14-13$, then $(a,b)=(1,-1)$ maps to $mub+nva=-14-13=-27 \pmod{91}$.
Section 2.6, page 22, in both paragraphs 2 and 3 of proof of Theorem 2.6.4:
$\gcd(n_1,n_2)$ not $\gcd(n_1n_2)$.
Section 2.6, 4 lines from bottom of page 22, $p_i^{e_i-1}(p_i-1)$, not
$p_i^{e_i-1}(p_1-1)$.
Chapter 3
Page 26, 4 lines from bottom: change "for all $p$" to "for all $a$".
Chapter 4
Proof of Theorem 4.3.1, page 31 (near the bottom of the page): in
the definition of $S$ change $\mathbb{R}^n$ to $\mathbb{R}^3$;
"Bye" should be "By" and in the last line $(x_0,y_0,z_0)$ should be
simply $(x,y,z)$.
Page 32, 2 lines after the end of the proof of Theorem 4.3.1: the
inequalities should be $|x|\le \sqrt{bc}$, $|y|\le \sqrt{ac}$,
$|z|\le \sqrt{ab}$.
Page 33, the end of the proof of Theorem 4.4.1 omits justification
that $\gcd(u,v)=1$: any common factor would divide both $x$ and $z$
but these are coprime by assumption.
Proof of Blichfled, top of page 36: change $S_2$ to $S$ three
times (i.e., in all place except the first one)
Chapter 5
Page 39, line 9: change "it natural" to "it is natural".
Page 41, line 5, proof of Thm. 5.3.4: $\alpha=p^m\varepsilon$
(exponent of $p$ should be $m$ not $n$).
Page 41, proof of Cor. 5.3.6, line 2: $d$ should be $b$.
Page 39, line 2 after (5.2.2), the second $x_2$ should be $x_3$.
Page 43, 4 lines from bottom: missing minus sign in front of $397/10$.
Page 44, Proof of Prop. 5.4.6, line 1: should say "From the proof
of Proposition 5.3.9" (not 5.4.4).
Page 44, Proof of Cor. 5.4.7, at the end: should say $\alpha=\lim
x_kp^m$ (i.e. multiply by $p^m$ not divide).
Page 46, last line: $\mathbb{Z}_2$, not $\mathbb{Z}_p$.