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# Inertial primes

Please do not rely on any information it contains.

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

For example, 2 is inertial in ${\displaystyle \mathbb {Z} [{\sqrt {5}}]}$ and ${\displaystyle \mathbb {Z} [{\sqrt {13}}]}$, as it is prime in both of those domains; but not in ${\displaystyle \mathbb {Z} [{\sqrt {-1}}]}$ or ${\displaystyle \mathbb {Z} [{\sqrt {2}}]}$ as it is composite in both: ${\displaystyle (1-i)(1+i)}$ in the former, ${\displaystyle ({\sqrt {2}})^{2}}$ in the latter.

If ${\displaystyle \mathbb {Z} [{\sqrt {d}}]}$ is a unique factorization domain and the Legendre symbol ${\displaystyle \left({\frac {d}{p}}\right)=-1}$, then ${\displaystyle p}$ is an inertial prime in ${\displaystyle \mathbb {Z} }$.[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? ${\displaystyle d}$ 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