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Arithmetic Tales pp 355–482Cite as

Algebraic Number Fields

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Abstract

This chapter is an introduction to algebraic number fields, which arose from both a generalization of the arithmetic in ℤ and the necessity to solve certain Diophantine equations. After recalling basic concepts from algebra and providing some polynomial irreducibility tools, the ring of integers \(\mathcal {O}_{\mathbb {K}}\) of an algebraic number field \(\mathbb {K}\) is investigated. Next, the ideal numbers, as Kummer called them, are introduced to restore unique factorization. The last section shows how analytic tools can be used to solve hard problems of algebraic number theory. In particular, an account of Zimmert’s method for ideal classes is given.

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Notes

  1. 1.

    The FLT states that the Fermat equation has no solution in positive integers x, y, z as soon as n⩾3. This was finally proved by Wiles in 1995.

  2. 2.

    For an interesting account of the history of the birth of the ideal theory in the late 19th century, the reader is referred to [Mol99, ST02].

  3. 3.

    See Sect. 2.6 in Chap. 2.

  4. 4.

    It should be mentioned that in this chapter we only consider unitary commutative rings unless explicitly stated to the contrary.

  5. 5.

    By Proposition 7.16 (v), all bases of a finitely generated abelian group have the same number of elements.

  6. 6.

    More generally, suppose we have a homomorphism \(\varphi: \mathbb {K}\longrightarrow \mathbb {L}\). Since these sets are fields, φ is injective and one may identify \(\mathbb {K}\) with its image \(\varphi(\mathbb {K})\), which is a subfield of \(\mathbb {L}\).

  7. 7.

    We sometimes make use of the equality P=P(X) where the right-hand side is the composition of the polynomial P with the polynomial X.

  8. 8.

    More generally, one can prove that if \(\mathbb {K}\) is a field such that \(\operatorname{char} \mathbb {K}= 0\), then \(P \in \mathbb {K}[X]\) can be written in the form P=Q 2 R if and only if P and P′ have a common factor of degree >0.

  9. 9.

    The curve y 2=P(x) is an example of hyperelliptic curve of genus 2 whose jacobian was first treated with a 2-descent method in [FPS97]. Using a refinement of a profound result due to Chabauty and Coleman, it may be shown that this curve has only six rational points, i.e. the two points at infinity and the points (0,±1) and (−3,±1).

  10. 10.

    This definition comes from the theory of resultants. The word “discriminant” indicates that \(\operatorname{disc}(P)\) does not vanish if all the roots α i are distinct so that \(\operatorname{disc}(P)\) discriminates the roots of P.

  11. 11.

    In a PID, every non-zero prime ideal is maximal. See [GG04, Théorème X.3.1] for instance.

  12. 12.

    Hermite, 1873. In 1882, using essentially the same ideas, Lindemann proved that π is also transcendental over ℚ and hence showed that squaring the circle is impossible.

  13. 13.

    This result is sometimes called the theorem of the primitive element.

  14. 14.

    See also [EM99] for another proof using the properties of sub-ℤ-modules of finitely generated ℤ-modules.

  15. 15.

    See Sect. 7.5 and the Kronecker–Weber theorem.

  16. 16.

    A subset of \(\mathbb {R}^{r_{1}+r_{2}}\) is discrete if and only if it intersects every closed ball of center O in a finite set.

  17. 17.

    A domain R is integrally closed means that if α/β is a root of a monic polynomial lying in R[X] with \(\alpha, \beta\in R, \ \beta\not= 0\), then α/β∈R.

  18. 18.

    In fact, we should rather denote these sets respectively by \(\mathcal{I} (\mathcal {O}_{\mathbb {K}})\) and \(\mathcal{P} (\mathcal {O}_{\mathbb {K}})\), but we have followed here the usual practice. Nevertheless, if R is any number ring, i.e. a domain for which the quotient field \(\mathbb {K}\) is an algebraic number field, then the collection of all fractional ideals, resp. principal ideals, of R is denoted by \(\mathcal{I}(R)\), resp. \(\mathcal{P} (R)\). Similarly, the class group or Picard group of R, which is an invariant of R, is denoted by Cl(R) or Pic(R) and is defined as in Definition 7.110 by \(\textrm{Cl}(R) = \mathcal{I}(R) / \mathcal {P} (R)\). When R is the ring of integers of an algebraic number field \(\mathbb {K}\), then \(\textrm{Cl}(\mathcal {O}_{\mathbb {K}})\) depends only on \(\mathbb {K}\) and is usually denoted by \(\operatorname{Cl(\mathbb {K})}\). Hence the class group in the usual sense may be viewed as an invariant of \(\mathbb {K}\).

  19. 19.

    The zero ideal can also be written as a product of prime ideals, but the decomposition is not unique. Hence the case of the zero ideal is almost always excluded in this book.

  20. 20.

    See footnote 18 for the notation.

  21. 21.

    The factor \(A_{\mathbb {K}}^{s} \Gamma_{\mathbb {K}}(s)\) is sometimes called the Euler factor of \(\zeta_{\mathbb {K}}(s)\).

  22. 22.

    See [Lan94, Nar04, Neu10] for an exhaustive account of Hecke’s method and Tate’s work.

  23. 23.

    Some authors [Coh93, Neu10] consider the “weighted” function \(2^{r_{2}}\xi_{\mathbb {K}} (s)\) instead.

  24. 24.

    A proof is supplied in Exercise 11.

  25. 25.

    The Extended Riemann Hypothesis states that all non-trivial zeros of the Dedekind zeta-function lie on the line σ=1/2. It should be mentioned that ERH is sometimes called GRH, for Generalized Riemann Hypothesis. However, GRH is referred to in this book as the conjecture asserting that the Dirichlet L-functions have all their zeros lying on the line σ=1/2. See Footnote 26.

  26. 26.

    The Generalized Riemann Hypothesis, or GRH for short, asserts that the Dirichlet L-functions have all their zeros lying on the line σ=1/2.

  27. 27.

    Implying in particular, along with a result of Ribet, Fermat’s last theorem.

  28. 28.

    See Definition 3.39.

  29. 29.

    This error remained unnoticed for about 90 years.

  30. 30.

    This is in fact the Hilbert class field of \(\mathbb {K}\). See Theorem 7.168.

  31. 31.

    This is now referred to as Kronecker Jurgendtraum i.e. Kronecker’s dream of his youth. This was completely proved by Takagi in 1920.

  32. 32.

    Among the 23 problems introduced by Hilbert, some of them have been solved and others are still open. Note that Hilbert’s 8th problem is the Riemann hypothesis.

  33. 33.

    For a definition of totally positive numbers in algebraic number fields, see [Nar04, page 44].

  34. 34.

    The different of \(\mathcal {O}_{\mathbb {K}}\) is the integral ideal \(\mathcal{D}_{\mathbb {K}/\mathbb {Q}}\) defined as the inverse of the fractional ideal

    $$\bigl\lbrace y \in \mathbb {K}: \textrm{Tr}_{\mathbb {K}/\mathbb {Q}}(xy) \in \mathbb {Z}\ \text{for all\ } x \in \mathcal {O}_\mathbb {K}\bigr\rbrace $$

    called the codifferent. It can be proved that the ramified prime ideals of \(\mathcal {O}_{\mathbb {K}}\) are the prime ideals dividing \(\mathcal{D}_{\mathbb {K}/\mathbb {Q}}\) and \(\mathcal {N}_{\mathbb {K}/\mathbb {Q}}( \mathcal{D}_{\mathbb {K}/\mathbb {Q}} ) = \vert d_{\mathbb {K}} \vert \). In fact, much more is true. Indeed, let p be a prime number and \(\mathfrak{p}\) be a prime ideal above p. If e is the ramification index \(e(\mathfrak{p} \mid p)\), then \(\mathfrak{p}^{e-1} \mid\mathcal {D}_{\mathbb {K}/\mathbb {Q}}\).

  35. 35.

    Such fields are usually called CM-fields.

  36. 36.

    PARI/GP provides \(\mathcal{R}_{\mathbb {K}}w_{\mathbb {K}}^{-1} \approx172 \, 495\).

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    Olivier Bordellès

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Bordellès, O. (2012). Algebraic Number Fields. In: Arithmetic Tales. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-4096-2_7

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