Inertial primes

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

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

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

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?	$d$	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