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Chapter 4: Problem 21

Let \(O\) be the ring of integers in a quadratic number field \(\mathbf{Q}(\sqrt{-d})\). Show that any subring \(R \subseteq O, R \neq \mathbf{Z},\) is of the form \(R=\mathbf{Z}+f \cdot O,\) for a uniquely deter mined integer \(f \geqslant 1 .\) This integer \(f\) is called the conductor of the ring \(R.\)

Short Answer

Expert verified
The subring \(R\) which is not equal to \(Z\) but lies in the set \(O\) is uniquely formulated by the positive integer \(f\). The integer \(f\) is the smallest positive non-zero element of the ring \(R\) not in \(Z\). By showing that \(f\) can only exist uniquely, it is referred to as the conductor of the subring \(R\).

Step by step solution

01

Start with the basic ring structure

Start with a set \(R\) which is a subring of \(O\) but not equal to \(Z\). This set is closed under addition, subtraction and multiplication.
02

Formulate the set \(R\)

Since \(R\) isn't equal to \(Z\), it must contain some element that isn't in \(Z\). Let \(f\) be the smallest non-zero positive integer that the subring \(R\) contains aside from \(Z\). We also can say that \(f\) belongs to \(O\), hence \(f\) can be expressed in the form of \(a+b\sqrt{-d}\) where \(a\) and \(b\) are elements within \(Z\).
03

Prove uniqueness of \(f\)

To prove the uniqueness of \(f\), assume there exists another integer \(g \geqslant 1\) such that \(R = \mathbf{Z} + g \cdot O\). Since \(f\leqslant g\) and \(f\) belongs to \(R\), we can write \(f = g + k\) for some integer \(k\). This gives us two possibilities, either \(k = 0\) or \(k \neq 0\). If \(k = 0\) then \(f = g\). If \(k \neq 0\), then \(g = f - k\) which contradicts the fact that \(g\) is an integer greater than or equal to 1 and \(f\) was chosen to be the smallest such integer. These facts combined leave us with \(f = g\) as our only option, hence \(f\) is unique.
04

Concluding discussion

The integer \(f\) is called the conductor of the ring \(R\) because it regulates the set up of the set \(R\) in the form described earlier, and it is unique for each subring. The smallest positive integer \(f\) that is in the ring but not in \(Z\) can only occur uniquely in the ring, thus making it the conductor of the ring in question.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ring of Integers
In the context of quadratic number fields, the ring of integers is an essential concept. A quadratic number field typically takes the form \(\mathbf{Q}(\sqrt{-d})\), where \(d\) is a positive integer.
The ring of integers within this field, denoted by \(O\), includes all numbers that can be expressed as \(a + b\sqrt{-d}\), where \(a\) and \(b\) are integers from the set \(\mathbf{Z}\).
  • It contains all numbers formed by the integer multiples of \(1\) and \(\sqrt{-d}\).
  • Rings of integers are closed under addition, subtraction, and multiplication.
  • This closure means any operation involving elements of \(O\) results in another element of \(O\).
Understanding the ring of integers helps us explore subrings and special properties like the conductor.
Subrings
Subrings are smaller sets contained within a larger ring that themselves satisfy the ring conditions. In our exercise, we are looking at subrings within the ring of integers \(O\) associated with \(\mathbf{Q}(\sqrt{-d})\).
For \(R\) to be a subring,
  • It must include the additive identity (zero).
  • It must be closed under addition and multiplication.
  • It must also be closed under subtraction.
If \(R\) is a subring of \(O\) but not equal to \(\mathbf{Z}\), it contains at least one element, aside from integers, that belongs to the ring \(O\), yet not to \(\mathbf{Z}\).
This element leads us into the next topic of conductors.
Conductor
The conductor is a critical integer in the study of these subrings. If \(R\) is expressed as \(R = \mathbf{Z} + f \cdot O\), then \(f\) is called the conductor. This integer \(f\) serves as a basis for constructing \(R\) from \(O\).
Here's what makes the conductor special:
  • It is the smallest positive integer that, along with \(\mathbf{Z}\), can recreate the subring \(R\).
  • The conductor determines how tightly the subring \(R\) is connected to the full ring \(O\).
Think of it as a bridge that allows \(R\) to form from \(\mathbf{Z}\) and \(O\).
Understanding the conductor helps in grasping the concept of uniqueness in these structures.
Uniqueness
Uniqueness refers to the distinct property possessed by the conductor within the structure of the subring \(R\). For any subring \(R\), there is a uniquely determined integer \(f\) that fits the requirement \(R = \mathbf{Z} + f \cdot O\).
To ensure this uniqueness:
  • Suppose another integer \(g \geqslant 1\) could also describe \(R\).
  • By comparing the sizes, with \(f\) being the smallest positive integer, mathematical contradictions show that no other integer except \(f\) forms \(R\).
Thus, \(f\) stands apart as the only integer that satisfies this role. The assurance of uniqueness plays a vital role because it prevents ambiguity in the formation and description of subrings.

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Most popular questions from this chapter

Prove the following lemma of Bertini: if \(X\) is a curve of degree \(d\) in \(\mathbf{P}^{3},\) not contained in any plane, then for almost all planes \(H \subseteq \mathbf{P}^{3}\) (meaning a Zariski open subset of the dual projective space \(\left(\mathbf{P}^{3}\right)^{*}\) ), the intersection \(X \cap H\) consists of exactly \(d\) distinct points, no three of which are collinear.

Show that a nonsingular plane curve of degree 5 has no \(g_{3}^{1}\). Show that there are nonhyperelliptic curves of genus 6 which cannot be represented as a nonsingular plane quintic curve.

Let \(X\) be an elliptic curve in \(\mathbf{P}^{2}\) given by an equation of the form \\[ y^{2}+\left(1, x y+u_{3} y=x^{3}+a_{2} x^{2}+u_{4} x+a_{6}\right. \\] Show that the \(j\) -invariant is a rational function of the \(c_{1},\) with coefficients in \(\mathbf{Q}\) In particular, if the \(a_{i}\) are all in some field \(k_{0} \subseteq k,\) then \(j \in k_{0}\) also. Furthermore for every \(x \in k_{0},\) there exists an elliptic curve defined over \(k_{0}\) with \(j\) -invariant equal to \(x\).

Let \(X\) be the curve \(y^{2}=x^{3}-7 x+10 .\) This curve has at least 26 points with integer coordinates. Find them (use a calculator), and verify that they are all contained in the subgroup (maybe equal to all of \(X(\mathbf{Q}) ?\) ) generated by \(P=(1,2)\) and \(Q=(2,2).\)

(a) Any automorphism of a curve of genus 3 is induced by an automorphism of \(\mathbf{P}^{2}\) via the canonical embedding. "(b) Assume char \(k \neq 3\). If \(X\) is the curve given by \\[ x^{3} y+y^{3} z+z^{3} x=0, \\] the group Aut \(X\) is the simple group of order 168 , whose order is the maximum \(84(g-1)\) allowed by (Ex. 2.5). See Burnside \([1, \$ 232]\) or Klein [1] "(c) Most curves of genus 3 have no automorphisms except the identity. [Hint: For each \(n,\) count the dimension of the family of curves with an automorphism \(T\) of order \(n\). For example, if \(n=2\), then for suitable choice of coordinates, \(T\) can be written as \(x \rightarrow-x, y \rightarrow y, z \rightarrow z .\) Then there is an 8 -dimensional family of curves fixed by \(T ;\) changing coordinates there is a 4 -dimensional family of such \(T\), so the curves having an automorphism of degree 2 form a family of dimensional 12 inside the 14 -dimensional family of all plane curves of degree \(4 .]\) Note: More generally it is true (at least over \(\mathbf{C}\) ) that for any \(g \geqslant 3\), a "sufficiently general" curve of genus \(g\) has no automorphisms except the identity-see Baily [1].
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