Kurt Hensel discovered or invented p-adic numbers around the end of the nineteenth century. This proves that there is a unique extension of the p-adic absolute value Qp to the algebraic closure Qp of Qp.
The Ring Zp of p-adic Integers
- Definition
- Addition of p-adic Integers
- The Ring of p-adic Integers
- The Order of a p-adic Integer
- Reduction mod p
- The Ring of p-adic Integers is a Principal Ideal Domain
In particular, all natural integers have an additive inverse in the set of p-adic integers. Another natural reason for this notation will appear in (3.6). The mapping a : Zy Zn obviously satisfies a 2 = or o or = id and is therefore an involution on the set of p-adic integers.
The Compact Space Zp
- Product Topology on Zp
- The Cantor Set
- Linear Models of Z P
- Free Monoids and Balls of Z P
- Euclidean Models
- An Exotic Example
By repeating the process, we obtain a descending order of nested compact subsets of the unit interval. Euclidean models of the ring of p-adic integers will be obtained in the next section by means of injective representations.
Topological Algebra
- Topological Groups
- Closed Subgroups of Topological Groups
- Quotients of Topological Groups
- Closed Subgroups of the Additive Real Line
- Closed Subgroups of the Additive Group of p-adic Integers
- Topological Rings
- Topological Fields, Valued Fields
A topological ring A is a ring with a topology such that A is an additive topological group and the multiplication is continuous on A x A. The topological group ZP is a complement to the additive group Z equipped with the induced topology.
Projective Limits 1. Introduction1.Introduction
- Definition
- Existence
- Projective Limits of Topological Spaces
- Projective Limits of Topological Groups
- Projective Limits of Topological Rings
- Back to the p-adic Integers
- Formal Power Series and p-adic Integers
Furthermore, if (E', >;) is another projective limit of the same sequence, there is a unique bijection f : E' -f E such that Y'n o f. The product of the K, is a compact space (Tychonoff's theorem), and the projective limit is a closed subspace of this compact space.
The Field Qp of p-adic Numbers
- The Fraction Field of Zp
- Ultrametric Structure on Q p
- Characterization of Rational Numbers Among p-adic Ones
- Fractional and Integral Parts of p-adic Numbers
- Additive Structure of Qp and Zp
- Euclidean Models of Q p
If the series (ai) is ultimately periodic, x is the sum of an integer and a linear combination (with integral coefficients) of series of the form. It defines an isomorphism Qp/Zp - p px of the additive group Qp/Z p with the group of pth roots of unity in the complex field C.
Hensel's Philosophy
- First Principle
- Algebraic Preliminaries
- Second Principle
- First Application: Invertible Elements in Zp
- Second Application: Square Roots in Qp
Similarly, we can find that the improvement x2 of the approximate root of x1 in p-adic integer form is satisfactory. Conversely, if a - 1 mod 8Z2, Hensel's lemma can be used to solve the equation X2 - a = 0, starting with the approximate solution x = 1.
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Fourth Application: Field Automorphisms of QP
It is possible to determine all automorphisms of the field QP (over the subfield Q). This shows that the algebraic automorphisms of the field Qp preserve p-adic order: they are automatically continuous. Let us emphasize that both in the p-adic and in the real case we are dealing with purely algebraic automorphisms over the subfield Q: The proofs show that they are automatically continuous and therefore trivial.
But there are infinitely many automorphisms of the complex field C: Only two of them are continuous, namely the identity and the complex conjugation. Alternatively, one can define the solenoid as the projective limit of the system with transition homomorphisms.
Torsion of the Solenoid
The kernel of this cover is of course ker VG = lim Z/pnZ = ZP, and we have the following short exact sequence of continuous homomorphisms,. Note that this unique cyclic subgroup C, of order m (prime to p) of Sp has a projection /i(Cm) into the circle given by. PROOF Let 6 : Z/pZ SP be any homomorphism of a cyclic group of order p in the solenoid.
Consequently, there is no element of order p in Sp (and even more so an element of order p' fork > 1 in Sr).
Embeddings of R and Q p in the Solenoid
The density of the image of follows from the density of the images of fn (1.4.4, Proposition 3) (in fact, all fn are surjective). In our context, take for A the connected subspace f (R) C SP which is dense in the solenoid. This follows immediately from the previous corollary, since the restriction of sum homomorphism to [0, 11 x Zp is already surjective, whereas its restriction to [0, 1) x Zp is bijective.
One must glue the two ends of the cylinder [0.11 x Zp with base Zp by a twist representing the unit shift of Zr. On the other hand, it is clear that instead of the subgroup r = Z[1/p] consisting of the elements (a, -a) (a E Z[1/p]) we could just as easily have used the diagonal subgroup A take , image of.
S. Closed Subgroups of the Solenoid
- Topological Properties of the Solenoid
- Ultrametric Spaces
Since H is compact, its image ip(H) is a closed subgroup of the circle R/Z. 1) The second is the easiest case. Show that the polynomial space K[X] compacts into K[[X]], so it is a completion of the polynomial space. Prove that r = Gd x (e) is a discrete subgroup of the topological group Gd x Go. a) Let H be a normal subgroup of the topological group G. Prove that the subgroup H is also normal.
Show that Gn form a fundamental system of identity neighborhoods in Gl2(Zp). A sphere of positive radius is open in an ultrametric space by part (c) of the previous lemma.
AMA CSC-AI
Absolute Values on Fields
An ultrametric field is a pair (K, 1.1) consisting of a field K and an ultrametric absolute value on K, namely an absolute value that satisfies the strong triangle inequality. Let x H Ix I be an absolute value on path K. 3) If x H 1xI is an ultrametric absolute value,.
Ultrametric Fields: The Representation Theorem
Since for any x E K there is an integer k such that I i; kx I < 1, namely such that i; kx E A, the preceding expansion can be derived for this element, and we obtain a series expansion for x starting at the index i = m = -k. Note that even when K is not complete, each x E K" has a series representation as indicated in the theorem, but an arbitrary series.
General Form of Hensel's Lemma
Since ICI' 0, the sequence is a Cauchy sequence and the series in>o aid i converges to the element x E A. Note that when the absolute value is trivial, taking only the values 0 and 1, the assumption reduces to .
Characterization of Ultrametric Absolute Values
The sequence (x, ),>o is a Cauchy sequence, so it converges in the complete field K. Since all iterates xi belong to the closed subring A, we have. Note that when the absolute value is trivial, it only takes the values 0 and 1, the assumption reduces to. iv) x H Ix I is an ultrametric absolute value. Indeed, any absolute value is bounded on the image of N in a field of characteristic p, since this image is a finite prime field.
The absolute values not bounded to the prime field of K (necessarily with characteristic zero) are sometimes called Archimedean absolute values: They have the property that. Distinct absolute values can define the same topology on a field K. It is not always useful to distinguish them.
Absolute Values on the Field Q
- Ultrametric Absolute Values on Q
- Generalized Absolute Values
- Ultrametric Among Generalized Absolute Values
- Generalized Absolute Values on the Rational Field
Note that if I .I is an absolute value and a > 0, then is not an absolute value in general. I is the usual absolute value on Q and a = 2, then f (x) = IX 12 does not satisfy the triangle inequality. If f is bounded on the image of the natural numbers N in K, then it is an ultrametric absolute value.
Every non-trivial generalized absolute value on a rational field Q is a power of an ordinary absolute value or a power of a p-adic absolute value. The previous result shows that for the generalized absolute value of f on the field Q are the only possibilities.
Finite-Dimensional Vector Spaces
- Nortned Spaces over Q p
- Locally Compact Vector Spaces over Q p
- Uniqueness of Extension of Absolute Values
- Existence of Extension of Absolute Values
- Locally Compact Ultrametric Fields
A norm defines an invariant (ultra-)metric on the underlying additive group of V. Hence a norm defines a topology on V, which becomes an additive topological group in which scalars are multiplied. absolute values of the elements of the field Qp. Consequently, this subspace L is closed and in the Hausdorff quotient V/L (1.3.3) the image A of the group 0 is a compact neighborhood of 0 and satisfies. There is at most one absolute value in K that extends the p-adic absolute value of Qp.
This observation can be used to prove the existence of the absolute value expansion of Qp. If a E Qp, it is obvious that N(a) = ad, whence IN(a)I11d = at, and the proposed formula is a p-adic value expansion.
Structure of p-adic Fields
- Degree and Residue Degree
- Totally Ramified Extensions
- Roots of Unity and Unramified Extensions
Since (si), a basis for the residual field k = R/P is considered a vector space over its prime field, we have. There can be no competition between the absolute values of the separated expressions x j n jan and this proves. There is no function defined in a region with z = 0 in C, so the field L' = L(zlle) is a proper extension of the field of convergent Laurent series L in the variable z.
In the course of the previous deduction, we again used the uniqueness of the expansion of estimates. Let K be a (commutative) field of characteristic 0 and let μ(K) be the multiplicative set consisting of roots of unity in K.
- Ramification and Roots of Unity
- Example 1: The Field of Gaussian 2-adic Numbers
- Example 2: The Hexagonal Field of 3-adic Numbers
- Example 3: A Composite of Totally Ramified Extensions Let us consider the following quadratic extensions of Q3
- Haar Measures
- Continuity of the Modulus
- Closed Balls are Compact
- The Modulus is a Strict Homomorphism
- S. Classification
- Finite-Dimensional Topological Vector Spaces
- Locally Compact Vector Spaces Revisited
- Final Comments on Regularity of Haar Measures
- Definition and First Properties
This proves that - when the residual field k is algebraic - the restriction of the reduction mod M is an isomorphism p(p)(K) -:; V. The residual field ka of the algebraic closure QP of Qp is an algebraic closure of the prime field F. The residual field of the maximum unbranched extension of Qp in Qp is an algebraic closure of the prime field F.
Let K be a finite extension of the p-adic field Qp and fix an algebraic element a c QP of degree n over K corresponding to a monic irreducible polynomial f E K[X] (of degree n). If L is an algebraic extension of K, then the residue field kL of L is also an algebraic extension of the residue field k of K. If K is a complete (non-discrete) ultrametric field and L is a finite extension of K, then at at most one extension of the absolute value of K to L.
We can proceed as in (1.5), since we now have uniqueness of the expansion of absolute values for finite extensions of K.
2.AEF,A'DA=A'EY
Ultrafilters
The inclusion relation for families FC P(X) is an order relation, and if .F' DF, we say that .F' is finer than Y. Every totally ordered set of filters on a set )), and according to Zorn's lemma, every filter is contained in a maximal filter.
Convergence and Compactness
Appendix to Chapter 3: Filters and Ultrafilters 155 For example, filtering neighborhoods of a point converges to this point. This proves that U contains an element of Y and this filter is finer than the filter of neighborhoods of Q. Let U be an ultrafilter on the compact space X and choose x in the non-empty intersection ,AEU A.
Let U be an ultrafilter on the set N of natural numbers and let (an )n>o be a bounded set of real numbers. The image of the ultrafilter U is a base of an ultrafilter in the compact space and therefore converges in this space.
Circular Filters
- Functions of an Integer Variable
- Integer-Valued Functions on the Natural Integers
- Integer-Valued Polynomial Functions
- Periodic Functions Taking Values in a Field of Characteristic pof Characteristic p
- Convolution of Functions of an Integer Variable
- Indefinite Sum of Functions of an Integer Variable
- Continuous Functions on Zp 1. Review of Some Classical Results Let us recall the basic property of uniform convergence
- Examples of p-adic Continuous Functions on Zp
- Mahler Series
- The Mahler Theorem
- Convolution of Continuous Functions on Zp
- Locally Constant Functions on Zp 1. Review of General Properties1.Review of General Properties
- Characteristic Functions of Balls of Zp
- The van der Put Theorem
- Ultrametric Banach Spaces
The aim of this chapter is the study of continuous functions on subsets of the p-adic field Qp with values in an extension of Qp. The action of the finite difference operator on the binomial functions is easy to determine: an elementary calculation shows this. It is sufficient to check this property for the basis of L consisting of the binomial polynomials.
It will be enough to check that the image of the set of binomial polynomials () is the canonical basis of the target. Another application of the Mahler theorem (or of the possibility of extending the convolution product to continuous functions over Zp) is given by the following corollary.
