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# abstract

### before second test

Question | Answer |
---|---|

additive identity | a+o=a |

multiplicative identity | ax=1 |

additive inverse | a+x=o |

multiplicative inverse | [a][n]=1 |

subring | ring closed under multiplication and subtraction |

commutative ring | xy=yx |

unity | multiplicative identity |

units | ab=ba=1 (a is a unit) |

zero divisor | R is a commutative ring, a does not equal 0, b does not equal 0, ab=0 |

integral domain | commutative ring but has no zero divisors |

field | commutative ring with unity in which every non-zero element is a unit |

factor of a polynomial | something you can factor out with remainder 0 |

root of a polynomial | plug in that number and get 0 |

Theorem 8.1 | Suppose R is an integral domain and a,b,c are elements of R with a not equal to 0. If ab=ac, then b=c |

Theorem 8.2 | A field has no zero divisors |

Theorem 8.5 | Zn is a field iff n is prime |

Theorem 8.6 | Let o less than x less than m. Then [x] is a unit in the ring Zm iff gcd (x,m)=1 |

Theorem 8.8 | All finite domains are fields |

examples of finite fields | Z5, Z3, Z2 |

examples of infinite field | Q |

finite integral domain that's not a field | impossible |

infinite integral domain that's not a field | Z |

a set that is closed under multiplication but not subtraction | N |

field without unity | impossible |

ring without unity | 2Z (even integers) |

polynomial in Q[x], reducible in Q[x] but has no roots in Q | x^4 -4 |

Created by:
monkeyhaha3