Download Full PDF Package. A short summary of this paper. Contributions to Algebraic Number Theory from India. Books – Erratum for Cassels-Froehlich – MathOverflow. Thanks to everyone who helped. Sign up using Email and Password. Pageline 2 after the diagram: For prerequisites, one could look at Milne’s notes on algebraic number theory, cassles first two chapters of Neukirch’s Algebraic Number Theoryor the first two chapters of Cassels-Frohlich, Algebraic Number Theory. Extension of the functor AG. For by a basic theorem of homological algebra, the H G, A) so defined satisfy the exactness property (1.3); also A = Hom (A, X) where X is an abelian group, then for any G-module B we have (the isomorphism being as follows: if: B -+ A is a G-homomorphism, then corresponds to the map B -4 X defined by b — where 1 is.

In mathematics, the ring of integers of an algebraic number fieldK is the ring of all integral elements contained in K. An integral element is a root of a monic polynomial with integer coefficients, xⁿ + c_n−1xⁿ⁻¹ + ... + c₀. This ring is often denoted by O_K or ${\displaystyle {\mathcal{O}}_K}$. Since any integer belongs to K and is an integral element of K, the ring Z is always a subring of O_K.

Prerequisites: Math 593 and 594 (PID’s, tensor products, and Galois theory including positive characteristic), and Math 575. Textbooks: There is no required text, but some books related to the course material will be kept on reserve at the library: Lang’s Algebraic Number Theory, Cassels-Fr olich’s Algebraic Number. Twin traps of doing algebra to the exclusion of number theory and of doing only trivial number theory. I take it for granted that the material I have chosen is interesting. My supreme stylistic goal is clarity.

The ring of integers Z is the simplest possible ring of integers.^[1] Namely, Z = O_Q where Q is the field of rational numbers.^[2] And indeed, in algebraic number theory the elements of Z are often called the 'rational integers' because of this.

The next simplest example is the ring of Gaussian integersZ[i], consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field Q(i) of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, it is a Euclidean domain.

The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain.^[3]

Properties[edit]

The ring of integers O_K is a finitely-generated Z-module. Indeed, it is a freeZ-module, and thus has an integral basis, that is a basisb₁, ... , b_n ∈ O_K of the Q-vector space K such that each element x in O_K can be uniquely represented as

${\displaystyle x=\sum _{i=1}^n{a}_i{b}_i,}$

with a_i ∈ Z.^[4] The rank n of O_K as a free Z-module is equal to the degree of K over Q.

Examples[edit]

Computational tool[edit]

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A useful tool for computing the integral close of the ring of integers in an algebraic field K/Q is using the discriminant. If K is of degree n over Q, and ${\displaystyle {\alpha }_1,\dots ,{\alpha }_n\in {\mathcal{O}}_K}$ form a basis of K over Q, set ${\displaystyle d={\mathrm{\Delta }}_{K/\mathbb{Q}}({\alpha }_1,\dots ,{\alpha }_n)}$. Then, ${\displaystyle {\mathcal{O}}_K}$ is a submodule of the Z-module spanned by ${\displaystyle {\alpha }_1/d,\dots ,{\alpha }_n/d}$^{[5]pg. 33}. In fact, if d is square-free, then this forms an integral basis for ${\displaystyle {\mathcal{O}}_K}$^{[5]pg. 35}.

Cyclotomic extensions[edit]

If p is a prime, ζ is a pth root of unity and K = Q(ζ) is the corresponding cyclotomic field, then an integral basis of O_K = Z[ζ] is given by (1, ζ, ζ², ... , ζ^p−2).^[6]

Quadratic extensions[edit]

If ${\displaystyle d}$ is a square-free integer and ${\displaystyle K=\mathbb{Q}(\sqrt{d})}$ is the corresponding quadratic field, then ${\displaystyle {\mathcal{O}}_K}$ is a ring of quadratic integers and its integral basis is given by (1, (1 + √d)/2) if d ≡ 1 (mod 4) and by (1, √d) if d ≡ 2, 3 (mod 4).^[7] This can be found by computing the minimal polynomial of an arbitrary element ${\displaystyle a+b\sqrt{d}\in \mathbf{Q}(\sqrt{d})}$ where ${\displaystyle a,b\in \mathbf{Z}}$.

Multiplicative structure[edit]

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In a ring of integers, every element has a factorization into irreducible elements, but the ring need not have the property of unique factorization: for example, in the ring of integers Z[√−5], the element 6 has two essentially different factorizations into irreducibles:^[3][8]

${\displaystyle 6=2\cdot 3=(1+\sqrt{-5})(1-\sqrt{-5}).}$

A ring of integers is always a Dedekind domain, and so has unique factorization of ideals into prime ideals.^[9]

The units of a ring of integers O_K is a finitely generated abelian group by Dirichlet's unit theorem. The torsion subgroup consists of the roots of unity of K. A set of torsion-free generators is called a set of fundamental units.^[10]

Generalization[edit]

One defines the ring of integers of a non-archimedean local fieldF as the set of all elements of F with absolute value ≤ 1; this is a ring because of the strong triangle inequality.^[11] If F is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers. The ring of integers of an algebraic number field may be characterised as the elements which are integers in every non-archimedean completion.^[2]

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For example, the p-adic integers Z_p are the ring of integers of the p-adic numbersQ_p.

See also[edit]

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• Integral closure – gives a technique for computing integral closures

References[edit]

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• Cassels, J.W.S. (1986). Local fields. London Mathematical Society Student Texts. 3. Cambridge: Cambridge University Press. ISBN0-521-31525-5. Zbl0595.12006.
• Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin: Springer-Verlag. ISBN978-3-540-65399-8. MR1697859. Zbl0956.11021.
• Samuel, Pierre (1972). Algebraic number theory. Hermann/Kershaw.

Notes[edit]

1. ^The ring of integers, without specifying the field, refers to the ring Z of 'ordinary' integers, the prototypical object for all those rings. It is a consequence of the ambiguity of the word 'integer' in abstract algebra.
2. ^ ^abCassels (1986) p. 192
3. ^ ^abSamuel (1972) p.49
4. ^Cassels (1986) p. 193
5. ^ ^abBaker. 'Algebraic Number Theory'(PDF). pp. 33–35.
6. ^Samuel (1972) p.43
7. ^Samuel (1972) p.35
8. ^Artin, Michael (2011). Algebra. Prentice Hall. p. 360. ISBN978-0-13-241377-0.
9. ^Samuel (1972) p.50
10. ^Samuel (1972) pp. 59–62
11. ^Cassels (1986) p. 41