First-Degree Prime Ideals of Biquadratic Fields dividing prescribed Principal Ideals

We describe first-degree prime ideals of biquadratic extensions in terms of first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms. Interestingly, the correspondence between these ideals in the larger ring and those in the smaller ones extends to the divisibility of principal ideals in their respective rings, with some exceptions that we explicitly provide. Finally, we hint at possible applications of this correspondence.


Introduction
Biquadratic fields are numerical fields that have been studied extensively [1,5,11,13,14,15] and are currently in the spotlight as they provide examples of nonprincipal Euclidean ideal classes [3,6]. They are defined as numerical fields whose Galois group is the Klein group and they can be obtained by the compositum of two quadratic fields, which is the construction we adopt in the present work.
In a general number field Q(γ), the special subring Z[γ] of the ring of integers plays an important role in applications, since their elements have a natural representation as integer-valued polynomials. Among its main properties, identifying its first-degree prime ideals is relatively easy as well as deciding whether they divide a given principal ideal [2]. This feature is crucial for the effectiveness of the General Number Field Sieve, which is the most efficient known algorithm to factorize large integers [7,8,9].
In this paper, we start from a biquadratic field Q(γ) and we investigate the relation between the first-degree prime ideals in Z[γ] and the first-degree prime ideals of Z[α] and Z[β], with Q(α) and Q(β) two underlying quadratic fields. In Section 2, we recall the results we need on the structure of first-degree prime ideals. In Section 3, we provide an explicit relation between such ideals in Z[γ] and those in Z[α] and Z[β], which depends on their ideal norms. In Section 4, we extend this relation to the divisibility of principal ideals in their respective rings, with some exceptions that we explicitly highlight. In Section 5, we hint at applications and further research.

Notation and preliminaries
Let Q(α) and Q(β) be two distinct quadratic fields, i.e. a = α 2 and b = β 2 have distinct non-trivial square-free parts. We may assume these number fields to be generated by the polynomials where a, b ∈ Z. It is well-known [14] that the biquadratic extension they generate is Q(γ) with γ = α + β, whose minimal polynomial is In this setting we focus on ideals in the order Z[θ], which has been deeply studied in [2]. Moreover, a non-zero prime ideal p of Z[θ] is called a first-degree prime ideal if N (p) is a prime integer.
The following theorem characterizes the first-degree prime ideals of Z[θ].
be an irreducible monic polynomial and θ ∈ C one of its roots. Then, for every positive prime p there is a bijection between In the above bijections an ideal p corresponds to (r, p) if it is the kernel of the evaluation-in-r ring morphism In the following sections we will use these bijections as identifications.
It is a standard fact [4] that non-zero ideals factor uniquely into primes in the whole ring of integers, however such a decomposition has been generalized even inside non-maximal orders Z[θ] in [2, Proposition 7.1]. The relation between these two types of factorization has been precisely characterized in [2,Proposition 7.2], and it founds a relevant employment for factoring in Z[θ] special principal ideals [2, Corollary 5.5], whose factorization only consists of first-degree prime ideals in the same order.

First-degree prime ideals of biquadratic extensions
The following theorem exhibits how to construct first-degree prime ideals of a given norm in a biquadratic field by knowing first-degree prime ideals of the same norm in two of its quadratic subfields. Proof. By hypothesis we have  Proof. We treat separately the cases p = 2 and p odd, explicitly exhibiting such a pair (r, s) in both cases.
Since in Z/2Z every element is equal to its square, the only choice of r, s ∈ Z/2Z satisfying • Case: p = 2 and t = 0.
In this case 2t is invertible in Z/pZ, then we can define . We notice that from f c (t) ≡ 0 mod p we have Thus, r 1 is a square root of a modulo p and since Z/pZ is a finite field there are at most two solutions to r 2 − a ≡ 0 mod p, hence these are Similarly, there are only two possible values for s, which are It is easy to verify that r 1 + s 1 = t and we now prove that (r 1 , s 1 ) is in fact the unique choice for such a pair (r, s) in order to satisfy r + s = t.
First, we notice that (r, s) = (r 2 , s 2 ) is not a possible option, since in this case r 2 + s 2 = −t but −t ≡ t mod p since p = 2 and t ≡ 0 mod p.
To conclude the proof we show that (r 1 , s 2 ) may be a suitable choice only when s 1 = s 2 = 0, therefore (r 1 , Since 2t 2 ≡ 0 mod p we get a + b ≡ t 2 mod p, then a ≡ t 2 mod p and b ≡ 0 mod p. This proves that s 1 = s 2 = 0. The same argument shows that (r 2 , s 1 ) is a valid pair only if r 1 = r 2 = 0 so (r 2 , s 1 ) = (r 1 , s 1 ). In conclusion, there is only one working pair, that is (r, s) = (r 1 , s 1 ).
The uniqueness part of Theorem 3.2 states that any ideal (t, p) of Z[γ] with t = 0 may be determined without repetitions from first-degree prime ideals of two underlying quadratic fields. The only ideals left are those of the form (0, p) for p = 2, which are examined in the following proposition. Proof. From f c (0) ≡ 0 mod p we get Hence, f a ≡ f b mod p, therefore r is a root of f a modulo p if and only if the same holds for f b . Moreover, if f a (r) ≡ 0 mod p also f a (−r) ≡ 0 mod p, implying that (±r, p) are first-degree prime ideals of Z[α], while the converse is trivial.
With the following theorems we prove that divisibility is stable under combination except for an exceptional case.
Proof. By Theorem 3.1 (r + s, p) is a first-degree prime ideal of Z[γ], then it is sufficient to show that under the aforementioned conditions we obtain (2) n + m(r + s) ≡ 0 mod p, which proves that I = n + mγ ⊆ ker π γ = (r + s, p), hence (r + s, p)|I.
Thus, either (2) is satisfied or π α (g) ≡ 1 mod 2, which implies that there are no first-degree prime ideals (r, 2) dividing I a .
In this case (2) is trivially satisfied.
Thus, we conclude that (r + s, p) is a first-degree prime ideal of Z[γ] dividing I except for the case p = 2, n ≡ 0 mod p and r + s ≡ 0 mod p.
The next example shows that in the exceptional case mentioned above ideal combination may not maintain divisibility.
On the other hand, whenever a first-degree prime ideal of Z[γ] dividing a given principal ideal I is obtained as a combination of two first-degree prime ideals, they divide the intersections of I with Z[α] and Z[β]. Proof. If these ideals exist, from (t, p) | I we get 0 ≡ n + mt ≡ n + mr + ms mod p. Let g a = n 2 + m 2 (a − b) + 2nmα a generator of I a and g b = n 2 + m 2 (b − a) + 2nmβ a generator of I b . From the above equation we have π α (g a ) = n 2 + m 2 (a − b) + 2nmr ≡ n 2 + m 2 (a − b) + 2n(−n − ms) mod p Hence, since π α (g a ) vanishes if and only if π β (g b ) does, it is sufficient to show that (r, p) | I a . Substituting n = −mr − ms, r 2 ≡ a mod p and s 2 ≡ b mod p we get π α (g a ) ≡ (−mr − ms) 2 + m 2 (r 2 − s 2 ) + 2(−mr − ms)mr ≡ 0 mod p, therefore (r, p) | I a .
The following corollary enhances the previous result in the generic case.  Thus, all or none the pairs , so it is still true that the combination preserves divisibility, but in general not uniquely.

Applications and further work
In this work we show how two first-degree prime ideals in quadratic extensions may be combined to obtain a first-degree prime ideal in the corresponding biquadratic extension lying over them. In addition, this correspondence is proved to preserve the division of prescribed first-degree prime ideals, except for some sporadic, though well-determined, cases.
Nonetheless, further computations suggest that our results might also be extended to more general number fields, possibly requiring additional hypotheses. Such a generalization could be repeatedly applied in order to characterize firstdegree prime ideals of a given norm in large extensions: the (at most) d 1 d 2 firstdegree prime ideals of a composite extension may be seen as combinations of d 1 +d 2 such ideals into smaller subrings, which are much more convenient to be stored and managed.
Among other applications, first-degree prime ideals are employed in the General Number Field Sieve, where a large number of them is needed to factorize some principal ideals. Even if biquadratic extensions may not be optimal for this algorithm [10,12], a more general form of our results on ideal combination could lead to better computational performance.
Finally, from a theoretical point of view, it may be worthy to investigate which algebraic properties are preserved, as happens for divisibility, by first-degree prime ideals combination.