Recommendation for Number Theory Textbook

Question

I'm a foreigner (meaning English is not my first language) and an undergraduate student. I'm currently studying linear algebra, set theory and have already studied number theory, and got a very good grade on the class!

However, the textbook we used on that number theory class was very poor (what a shame...). I used to love number theory, so the only obstacle for that class was that poor textbook. Now, even though I got good grade on that class, my brain is messed up with all the number theory stuff - it is not well organized, so I want you to recommend me a suitable textbook.

The conditions are (1) I have to review the stuff so that all the theorems should be well organized in my brain (2) Since I've already took the class, I want a little bit more challenge. The problem is that I haven't took that many classes such as analysis or other algebra. So I doubt if there is a suitable book that can satisfy both conditions.

I have already checked possible recommendations here. But there were just so many recommendations so I couldn't find which suits me best.

Answer 1

Take a look at how to solve binary form $ax^2+bxy+cy^2=m$, for integer and rational $ (x,y)$ . It seems to me you are in good position to learn all the ingredients. I can also tell you that many students who take algebraic number theory have no idea how to solve such problems. So, my general feeling is that a course, maybe half-length, in integral binary quadratic forms, is a good lead-in to algebraic number theory.

And, since you know fields and matrices but not rings, it is probably a good choice at this point. Two books are involved, Buell and Conway. I have put quite a number of answers on this site in this style; sometimes the Conway topograph only, sometimes the Lagrange cycle method only...

Sample:

Pell x^2 -  12553 y^2.

0  form   1 224 -9   delta  -24
1  form   -9 208 193   delta  1
2  form   193 178 -24   delta  -8
3  form   -24 206 81   delta  2
4  form   81 118 -112   delta  -1
5  form   -112 106 87   delta  1
6  form   87 68 -131   delta  -1
7  form   -131 194 24   delta  8
8  form   24 190 -147   delta  -1
9  form   -147 104 67   delta  2
10  form   67 164 -87   delta  -2
11  form   -87 184 47   delta  4
12  form   47 192 -71   delta  -2
13  form   -71 92 147   delta  1
14  form   147 202 -16   delta  -13
15  form   -16 214 69   delta  3
16  form   69 200 -37   delta  -5
17  form   -37 170 144   delta  1
18  form   144 118 -63   delta  -2
19  form   -63 134 128   delta  1
20  form   128 122 -69   delta  -2
21  form   -69 154 96   delta  1
22  form   96 38 -127   delta  -1
23  form   -127 216 7   delta  31
24  form   7 218 -96   delta  -2
25  form   -96 166 59   delta  3
26  form   59 188 -63   delta  -3
27  form   -63 190 56   delta  3
28  form   56 146 -129   delta  -1
29  form   -129 112 73   delta  2
30  form   73 180 -61   delta  -3
31  form   -61 186 64   delta  3
32  form   64 198 -43   delta  -4
33  form   -43 146 168   delta  1
34  form   168 190 -21   delta  -9
35  form   -21 188 177   delta  1
36  form   177 166 -32   delta  -6
37  form   -32 218 21   delta  10
38  form   21 202 -112   delta  -1
39  form   -112 22 111   delta  1
40  form   111 200 -23   delta  -9
41  form   -23 214 48   delta  4
42  form   48 170 -111   delta  -1
43  form   -111 52 107   delta  1
44  form   107 162 -56   delta  -3
45  form   -56 174 89   delta  2
46  form   89 182 -48   delta  -4
47  form   -48 202 49   delta  4
48  form   49 190 -72   delta  -2
49  form   -72 98 141   delta  1
50  form   141 184 -29   delta  -7
51  form   -29 222 8   delta  27
52  form   8 210 -191   delta  -1
53  form   -191 172 27   delta  7
54  form   27 206 -72   delta  -2
55  form   -72 82 151   delta  1
56  form   151 220 -3   delta  -74
57  form   -3 224 3   delta  74
58  form   3 220 -151   delta  -1
59  form   -151 82 72   delta  2
60  form   72 206 -27   delta  -7
61  form   -27 172 191   delta  1
62  form   191 210 -8   delta  -27
63  form   -8 222 29   delta  7
64  form   29 184 -141   delta  -1
65  form   -141 98 72   delta  2
66  form   72 190 -49   delta  -4
67  form   -49 202 48   delta  4
68  form   48 182 -89   delta  -2
69  form   -89 174 56   delta  3
70  form   56 162 -107   delta  -1
71  form   -107 52 111   delta  1
72  form   111 170 -48   delta  -4
73  form   -48 214 23   delta  9
74  form   23 200 -111   delta  -1
75  form   -111 22 112   delta  1
76  form   112 202 -21   delta  -10
77  form   -21 218 32   delta  6
78  form   32 166 -177   delta  -1
79  form   -177 188 21   delta  9
80  form   21 190 -168   delta  -1
81  form   -168 146 43   delta  4
82  form   43 198 -64   delta  -3
83  form   -64 186 61   delta  3
84  form   61 180 -73   delta  -2
85  form   -73 112 129   delta  1
86  form   129 146 -56   delta  -3
87  form   -56 190 63   delta  3
88  form   63 188 -59   delta  -3
89  form   -59 166 96   delta  2
90  form   96 218 -7   delta  -31
91  form   -7 216 127   delta  1
92  form   127 38 -96   delta  -1
93  form   -96 154 69   delta  2
94  form   69 122 -128   delta  -1
95  form   -128 134 63   delta  2
96  form   63 118 -144   delta  -1
97  form   -144 170 37   delta  5
98  form   37 200 -69   delta  -3
99  form   -69 214 16   delta  13
100  form   16 202 -147   delta  -1
101  form   -147 92 71   delta  2
102  form   71 192 -47   delta  -4
103  form   -47 184 87   delta  2
104  form   87 164 -67   delta  -2
105  form   -67 104 147   delta  1
106  form   147 190 -24   delta  -8
107  form   -24 194 131   delta  1
108  form   131 68 -87   delta  -1
109  form   -87 106 112   delta  1
110  form   112 118 -81   delta  -2
111  form   -81 206 24   delta  8
112  form   24 178 -193   delta  -1
113  form   -193 208 9   delta  24
114  form   9 224 -1   delta  -224
115  form   -1 224 9   delta  24
116  form   9 208 -193   delta  -1
117  form   -193 178 24   delta  8
118  form   24 206 -81   delta  -2
119  form   -81 118 112   delta  1
120  form   112 106 -87   delta  -1
121  form   -87 68 131   delta  1
122  form   131 194 -24   delta  -8
123  form   -24 190 147   delta  1
124  form   147 104 -67   delta  -2
125  form   -67 164 87   delta  2
126  form   87 184 -47   delta  -4
127  form   -47 192 71   delta  2
128  form   71 92 -147   delta  -1
129  form   -147 202 16   delta  13
130  form   16 214 -69   delta  -3
131  form   -69 200 37   delta  5
132  form   37 170 -144   delta  -1
133  form   -144 118 63   delta  2
134  form   63 134 -128   delta  -1
135  form   -128 122 69   delta  2
136  form   69 154 -96   delta  -1
137  form   -96 38 127   delta  1
138  form   127 216 -7   delta  -31
139  form   -7 218 96   delta  2
140  form   96 166 -59   delta  -3
141  form   -59 188 63   delta  3
142  form   63 190 -56   delta  -3
143  form   -56 146 129   delta  1
144  form   129 112 -73   delta  -2
145  form   -73 180 61   delta  3
146  form   61 186 -64   delta  -3
147  form   -64 198 43   delta  4
148  form   43 146 -168   delta  -1
149  form   -168 190 21   delta  9
150  form   21 188 -177   delta  -1
151  form   -177 166 32   delta  6
152  form   32 218 -21   delta  -10
153  form   -21 202 112   delta  1
154  form   112 22 -111   delta  -1
155  form   -111 200 23   delta  9
156  form   23 214 -48   delta  -4
157  form   -48 170 111   delta  1
158  form   111 52 -107   delta  -1
159  form   -107 162 56   delta  3
160  form   56 174 -89   delta  -2
161  form   -89 182 48   delta  4
162  form   48 202 -49   delta  -4
163  form   -49 190 72   delta  2
164  form   72 98 -141   delta  -1
165  form   -141 184 29   delta  7
166  form   29 222 -8   delta  -27
167  form   -8 210 191   delta  1
168  form   191 172 -27   delta  -7
169  form   -27 206 72   delta  2
170  form   72 82 -151   delta  -1
171  form   -151 220 3   delta  74
172  form   3 224 -3   delta  -74
173  form   -3 220 151   delta  1
174  form   151 82 -72   delta  -2
175  form   -72 206 27   delta  7
176  form   27 172 -191   delta  -1
177  form   -191 210 8   delta  27
178  form   8 222 -29   delta  -7
179  form   -29 184 141   delta  1
180  form   141 98 -72   delta  -2
181  form   -72 190 49   delta  4
182  form   49 202 -48   delta  -4
183  form   -48 182 89   delta  2
184  form   89 174 -56   delta  -3
185  form   -56 162 107   delta  1
186  form   107 52 -111   delta  -1
187  form   -111 170 48   delta  4
188  form   48 214 -23   delta  -9
189  form   -23 200 111   delta  1
190  form   111 22 -112   delta  -1
191  form   -112 202 21   delta  10
192  form   21 218 -32   delta  -6
193  form   -32 166 177   delta  1
194  form   177 188 -21   delta  -9
195  form   -21 190 168   delta  1
196  form   168 146 -43   delta  -4
197  form   -43 198 64   delta  3
198  form   64 186 -61   delta  -3
199  form   -61 180 73   delta  2
200  form   73 112 -129   delta  -1
201  form   -129 146 56   delta  3
202  form   56 190 -63   delta  -3
203  form   -63 188 59   delta  3
204  form   59 166 -96   delta  -2
205  form   -96 218 7   delta  31
206  form   7 216 -127   delta  -1
207  form   -127 38 96   delta  1
208  form   96 154 -69   delta  -2
209  form   -69 122 128   delta  1
210  form   128 134 -63   delta  -2
211  form   -63 118 144   delta  1
212  form   144 170 -37   delta  -5
213  form   -37 200 69   delta  3
214  form   69 214 -16   delta  -13
215  form   -16 202 147   delta  1
216  form   147 92 -71   delta  -2
217  form   -71 192 47   delta  4
218  form   47 184 -87   delta  -2
219  form   -87 164 67   delta  2
220  form   67 104 -147   delta  -1
221  form   -147 190 24   delta  8
222  form   24 194 -131   delta  -1
223  form   -131 68 87   delta  1
224  form   87 106 -112   delta  -1
225  form   -112 118 81   delta  2
226  form   81 206 -24   delta  -8
227  form   -24 178 193   delta  1
228  form   193 208 -9   delta  -24
229  form   -9 224 1   delta  224
230  form   1 224 -9

 disc   50212
Automorph, written on right of Gram matrix:  
6569654811042976036551664287776695086862221675257028799978001629441955413114961078437403743404915644466809863876018081  1471866589689666239349905729201246719313821836636303758803979047726858179020247176803043556748067158933333371566607194904
163540732187740693261100636577916302145980204070700417644886560858539797668916352978115950749785239881481485729623021656  36639693664864958266523094257741028375786427933512150581254567633942356633250378028176410371695298649096319613299432869025


 Pell automorph 
18323131659838000621279822961014402535436645077593703805027272817785899294331746494627423887719351782370393211581654443553  2052926811152708922506596290962583340838489501699502342696260998457250080137906978934289529762054116232237090363957790847768
163540732187740693261100636577916302145980204070700417644886560858539797668916352978115950749785239881481485729623021656  18323131659838000621279822961014402535436645077593703805027272817785899294331746494627423887719351782370393211581654443553

Pell unit 
18323131659838000621279822961014402535436645077593703805027272817785899294331746494627423887719351782370393211581654443553^2 - 12553 * 163540732187740693261100636577916302145980204070700417644886560858539797668916352978115950749785239881481485729623021656^2 = 1 

=========================================

Pell NEGATIVE 
3026807861414232601576476947093931340883926646932707209616924^2 - 12553 * 27015380505739903416049453225901131788373049465835213058397^2 = -1 

=========================================

12553       12553

=========================

Answer 2

It wasn't clear to me whether you wanted a book that covered what you've already learnt (but in a clearer manner) or whether you want something to build on what you've learnt.

If it is the latter you want, then I recommend A classical introduction to modern number theory, by Ireland and Rosen. It is a beautiful book, that begins with equations mod $p$, Gauss sums, and related (fairly elementary) topics, and builds up to a treatment of the basic theorems from algebraic number theory (but in a manner which requires essentially no prior algebra background, if my memory is correct).