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# Inertial primes

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Given a quadratic integer ring , its **inertial primes** are those prime numbers in (this includes the positive primes of A000040 and those primes multiplied by –1) that are also prime in . The term is a contrast for the terms for primes in that are composite in , which "ramify" or "split" depending on whether the factorization involves the square of another prime.

For example, 2 is inertial in and , as it is prime in both of those domains; but not in or as it is composite in both: in the former, in the latter.

If is a unique factorization domain and the Legendre symbol , then is an inertial prime in .^{[1]}

## Table of inertial primes in some imaginary fields

In the following table, P means inertial prime, ^ means the square of a prime with nonzero imaginary part and * means the product of a prime with nonzero imaginary part and one of its associates.

TABLE GOES HERE

## Table of inertial primes in some real fields

In the following table, P means inertial prime, ^ means the square of a prime with nonzero "radical" part and * means the product of a prime with nonzero "radical" part and one of its associates (the factorization may include the unit –1).

UFD? | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | ||

✓ | 2 | ^ | P | P | * | P | P | * | P | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |

✓ | 3 | * | ^ | P | P | * | * | P | P | * | P | P | * | P | P | * | A003630 |

✓ | 5 | P | P | ^ | P | * | P | P | * | P | * | * | P | * | P | P | A003631 |

✓ | 6 | * | * | * | P | P | P | P | * | * | * | P | P | P | * | * | A038877 |

✓ | 7 | * | * | P | ^ | P | P | P | * | P | * | * | * | P | P | * | A003632 |

✗ | 10 | P | P | P | P | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |

✓ | 11 | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |

✓ | 13 | P | * | P | P | P | ^ | * | P | * | * | P | P | P | * | P | A038884 |

✓ | 14 | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |

✗ | 15 | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |

✓ | 17 | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |

✓ | 19 | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |

- Bolker, p. 107

- ↑ See Theorem P3 in Quadratic integer rings#Primes in quadratic integer rings.