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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?23571113171923293137414347
2^PP*PP*P23293137414347
3*^PP**PP*PP*PP*A003630
5PP^P*PP*P**P*PPA003631
6***PPPP***PPP**A038877
7**P^PPP*P***PP*A003632
10PPPP1113171923293137414347
1123571113171923293137414347
13P*PPP^*P**PPP*PA038884
1423571113171923293137414347
1523571113171923293137414347
1723571113171923293137414347
1923571113171923293137414347
  • Bolker, p. 107