Notations |
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xiii | |
Preface |
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xvii | |
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1 | (26) |
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1 | (4) |
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5 | (5) |
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10 | (2) |
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12 | (3) |
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15 | (3) |
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18 | (3) |
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21 | (6) |
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27 | (12) |
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27 | (3) |
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30 | (8) |
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Bibliography of Niels Abel |
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38 | (1) |
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39 | (20) |
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Definition and Examples of Groups |
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39 | (8) |
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Elementary Properties of Groups |
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47 | (3) |
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50 | (2) |
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52 | (7) |
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3 Finite Groups; Subgroups |
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59 | (18) |
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59 | (2) |
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61 | (3) |
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64 | (5) |
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69 | (8) |
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77 | (20) |
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Properties of Cyclic Groups |
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77 | (6) |
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Classification of Subgroups of Cyclic Groups |
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83 | (13) |
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Bibliography of James Joseph Sylvester |
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96 | (1) |
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97 | (30) |
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97 | (3) |
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100 | (3) |
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Properties of Permutations |
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103 | (12) |
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A Check-Digit Scheme Based on D5 |
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115 | (10) |
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Bibliography of Augustin Cauchy |
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125 | (1) |
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Bibliography of Alan Taring |
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126 | (1) |
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127 | (22) |
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127 | (1) |
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127 | (4) |
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Properties of Isomorphisms |
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131 | (3) |
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134 | (4) |
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138 | (3) |
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141 | (6) |
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Bibliography of Arthur Cayley |
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147 | (2) |
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7 Cosets And Lagrange's Theorem |
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149 | (22) |
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149 | (4) |
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Lagrange's Theorem and Consequences |
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153 | (5) |
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An Application of Cosets to Permutation Groups |
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158 | (2) |
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The Rotation Group of a Cube and a Soccer Ball |
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160 | (3) |
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An Application of Cosets to the Rubik's Cube |
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163 | (1) |
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163 | (7) |
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Bibliography of Joseph Lagrange |
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170 | (1) |
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8 External Direct Products |
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171 | (22) |
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171 | (2) |
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Properties of External Direct Products |
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173 | (3) |
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The Group of Units Modulo n as an External Direct Product |
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176 | (2) |
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178 | (6) |
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184 | (7) |
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Bibliography of Leonard Adleman |
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191 | (2) |
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9 Normal Subgroups And Factor Groups |
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193 | (26) |
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193 | (3) |
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196 | (4) |
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Applications of Factor Groups |
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200 | (3) |
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203 | (6) |
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209 | |
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Bibliography of Evariste Galois |
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204 | (15) |
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219 | (22) |
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219 | (2) |
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Properties of Homomorphisms |
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221 | (4) |
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The First Isomorphism Theorem |
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225 | (7) |
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232 | (7) |
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Bibliography of Camille Jordan |
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239 | (2) |
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11 Fundamental Theorem Of Finite Abelian Groups |
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241 | (14) |
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241 | (1) |
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The Isomorphism Classes of Abelian Groups |
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242 | (4) |
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Proof of the Fundamental Theorem |
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246 | (3) |
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249 | (6) |
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255 | (12) |
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Motivation and Definition |
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255 | (1) |
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256 | (1) |
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257 | (2) |
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259 | (2) |
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261 | (5) |
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Bibliography of I. N. Herstein |
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266 | (1) |
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267 | (12) |
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267 | (1) |
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268 | (3) |
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271 | (2) |
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273 | (6) |
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14 Ideals And Factor Rings |
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279 | (16) |
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279 | (1) |
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280 | (4) |
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Prime Ideals and Maximal Ideals |
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284 | (2) |
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286 | (7) |
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Bibliography of Richard Dedekind |
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293 | (1) |
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Bibliography of Emmy Noether |
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294 | (1) |
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295 | (16) |
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295 | (3) |
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Properties of Ring Homomorphisms |
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298 | (3) |
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301 | (2) |
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303 | (8) |
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311 | (14) |
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311 | (3) |
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The Division Algorithm and Consequences |
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314 | (5) |
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319 | (6) |
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17 Factorization Of Polynomials |
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325 | (18) |
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325 | (3) |
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328 | (6) |
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Unique Factorization in Z[ x] |
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334 | (1) |
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Weird Dice: An Application of Unique Factorization |
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335 | (3) |
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338 | (4) |
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Bibliography of Serge Lang |
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342 | (1) |
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18 Divisibility In Integral Domains |
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343 | (22) |
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343 | (3) |
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Historical Discussion of Fermat's Last Theorem |
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346 | (4) |
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Unique Factorization Domains |
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350 | (3) |
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353 | (3) |
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356 | (5) |
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Bibliography of Sophie Germain |
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361 | (1) |
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Bibliography of Andrew Wiles |
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362 | (1) |
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Bibliography of Pierre de Fermat |
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363 | (2) |
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365 | (20) |
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The Fundamental Theorem of Field Theory |
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365 | (2) |
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367 | (7) |
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Zeros of an Irreducible Polynomial |
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374 | (5) |
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379 | (4) |
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Bibliography of Leopold Kronecker |
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383 | (2) |
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385 | (16) |
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Characterization of Extensions |
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385 | (2) |
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387 | (5) |
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Properties of Algebraic Extensions |
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392 | (2) |
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394 | (6) |
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Bibliography of Ernst Steinitz |
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400 | (1) |
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401 | (18) |
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Classification of Finite Fields |
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401 | (1) |
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Structure of Finite Fields |
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402 | (6) |
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Subfields of a Finite Field |
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408 | (2) |
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410 | (6) |
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Bibliography of L. E. Dickson |
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416 | (1) |
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Bibliography of E. H. Moore |
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417 | (2) |
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22 Geometric Constructions |
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419 | (8) |
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Historical Discussion of Geometric Constructions |
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419 | (1) |
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420 | (2) |
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Angle-Trisectors and Circle-Squarers |
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422 | (1) |
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423 | (4) |
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427 | (20) |
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427 | (1) |
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428 | (2) |
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430 | (6) |
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Applications of Sylow Theorems |
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436 | (4) |
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440 | (5) |
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Bibliography of Ludwig Sylow |
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445 | (2) |
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447 | (20) |
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447 | (6) |
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453 | (5) |
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458 | (2) |
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460 | (1) |
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461 | (1) |
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461 | (3) |
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Bibliography of Michael Aschbacher |
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464 | (1) |
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Bibliography of Daniel Gorenstein |
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465 | (1) |
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Bibliography of John Thompson |
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466 | (1) |
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25 Generators And Relations |
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467 | (16) |
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467 | (1) |
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468 | (1) |
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469 | (2) |
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471 | (4) |
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Classification of Groups of Order Up to 15 |
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475 | (1) |
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Characterization of Dihedral Groups |
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476 | (3) |
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479 | (3) |
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Bibliography of Marshall Hall, Jr. |
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482 | (1) |
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483 | (8) |
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483 | (2) |
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Classification of Finite Plane Symmetry Groups |
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485 | (2) |
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Classification of Finite Groups of Rotations in R3 |
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487 | (1) |
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488 | (3) |
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491 | (12) |
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491 | (1) |
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492 | (2) |
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494 | (4) |
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498 | (1) |
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499 | (2) |
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Bibliography of William Burnside |
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501 | (2) |
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28 Cayley Digraphs Of Groups |
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503 | (22) |
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503 | (1) |
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The Cayley Digraph of a Group |
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503 | (4) |
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Hamiltonian Circuits and Paths |
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507 | (7) |
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514 | (4) |
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518 | (4) |
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Bibliography of William Rowan Hamilton |
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522 | (1) |
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Bibliography of Paul Erdos |
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523 | (2) |
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29 Introduction To Algebraic Coding Theory |
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525 | (30) |
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525 | (5) |
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530 | (6) |
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Parity-Check Matrix Decoding |
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536 | (3) |
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539 | (4) |
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543 | (3) |
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546 | (5) |
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Bibliography of Richard W. Hamming |
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551 | (1) |
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Bibliography of Jessie Mac Williams |
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552 | (1) |
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Bibliography of Vera Pless |
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553 | (2) |
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30 An Introduction To Galois Theory |
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555 | (18) |
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Fundamental Theorem of Galois Theory |
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555 | (7) |
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Solvability of Polynomials by Radicals |
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562 | (6) |
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Insolvability of a Quintic |
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568 | (1) |
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569 | (4) |
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573 | (12) |
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573 | (1) |
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574 | (5) |
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The Constructible Regular n-gons |
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579 | (2) |
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581 | (2) |
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Bibliography of Carl Friedrich Gauss |
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583 | (1) |
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Bibliography of Manjul Bhargava |
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584 | (1) |
Selected Answers |
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585 | (44) |
Index |
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629 | |