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Formally real field

A field $F$ which is capable of being made an ordered field. The existence of such an order is equivalent to the property that $-1$ is not a sum of squares in $F$: this is the Artin--Schreier theorem. A real closed field is a formally real field for which no algebraic extension is formally real.

References

• Tsit Yuen Lam, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, American Mathematical Society (2005) ISBN 0-8218-1095-2 Zbl 1068.11023 MR2104929
• A. R. Rajwade, Squares, London Mathematical Society Lecture Note Series 171 Cambridge University Press (1993) ISBN 0-521-42668-5 Zbl 0785.11022
• J.W. Milnor, D. Husemöller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete 73, Springer-Verlag (1973) ISBN 0-387-06009-X Zbl 0292.10016

Quadratically closed field

A field in which every element of the field has a square root in the field.

Examples

• The field of complex numbers is quadratically closed; more generally, any algebraically closed field is quadratically closed.
• The field of real numbers is not quadratically closed as it does not contain a square root of $-1$.
• The union of the finite fields $F_{5^{2^n}}$ for $n \ge 0$ is quadratically closed but not algebraically closed.
• The field of constructible numbers is quadratically closed but not algebraically closed.

Properties

• A field is quadratically closed if and only if it has universal invariant equal to 1.
• Every quadratically closed field is a Pythagorean field but not conversely (for example, $\mathbb{R}$ is Pythagorean); however, every non-formally real Pythagorean field is quadratically closed.
• A field is quadratically closed if and only if its Witt–Grothendieck ring is isomorphic to $\mathbb{Z}$ under the dimension mapping.
• A formally real Euclidean field $E$ is not quadratically closed (as $-1$ is not a square in $E$) but the quadratic extension $E(\sqrt{-1})$ is quadratically closed.
• Let $E/F$ be a finite extension where $E$ is quadratically closed. Either $-1$ is a square in $F$ and $F$ is quadratically closed, or $-1$ is not a square in $F$ and $F$ is Euclidean. This "going-down theorem" may be deduced from the Diller–Dress theorem.

Quadratic closure

A quadratic closure of a field $F$ is a quadratically closed field which embeds in any other quadratically closed field containing $F$. A quadratic closure for a given $F$ may be constructed as a subfield of the algebraic closure $F^{\mathrm{alg}}$ of $F$, as the union of all quadratic extensions of $F$ in $F^{\mathrm{alg}}$.

Examples

• The quadratic closure of the field of real numbers is the field of complex numbers.
• The quadratic closure of the finite field $\mathbb{F}_5$ is the union of the $\mathbb{F}_{5^{2^n}}$.
• The quadratic closure of the field of rational numbers is the field of constructible numbers.

References

• Tsit Yuen Lam, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, American Mathematical Society (2005) ISBN 0-8218-1095-2 Zbl 1068.11023 MR2104929
• A. R. Rajwade, Squares, London Mathematical Society Lecture Note Series 171 Cambridge University Press (1993) ISBN 0-521-42668-5 Zbl 0785.11022

Baer–Specker group

An example of an infinite Abelian group which is a building block in the structure theory of such groups.

Definition

The Baer-Specker group is the group B = Z^N of all integer sequences with componentwise addition, that is, the direct product of countably many copies of Z.

Properties

Reinhold Baer proved in 1937 that this group is not free abelian; Specker proved in 1950 that every countable subgroup of B is free abelian.

See also

• Slender group

References

• Phillip A. Griffith Infinite Abelian group theory, Chicago Lectures in Mathematics, University of Chicago Press (1970) ISBN 0-226-30870-7. pp.1, 111-112

Cameron–Erdős conjecture

The Cameron-Erdős conjecture in the field of combinatorics within mathematics is the statement that the number of sum-free sets contained in \(\{1,\ldots,N\}\) is \(O\left({2^{N/2}}\right)\).

The conjecture was stated by Peter Cameron and Paul Erdős in 1988. It was proved by Ben Green in 2003 .

Conductor of an abelian variety

In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is. It is connected to the ramification in the field generated by the division points.

For an abelian variety A defined over a field F with ring of integers R, consider the Néron model of A, which is a 'best possible' model of A defined over R. This model may be represented as a scheme over

Spec(R)

(cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism

Spec(F) → Spec(R)

gives back A. Let A⁰ denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a residue field k, A⁰_k is a group variety over k, hence an extension of an abelian variety by a linear group. This linear group is an extension of a torus by a unipotent group. Let u be the dimension of the unipotent group and t the dimension of the torus. The order of the conductor is

\[ f = 2u + t + \delta , \, \]

where δ is a measure of wild ramification.

Properties

• If A has good reduction then f = u = t = δ = 0.
• If A has semistable reduction or, more generally, acquires semistable reduction over a Galois extension of F of degree prime to p, the residue characteristic, then δ = 0.
• If p > 2d + 1, where d is the dimension of A, then δ = 0.

References

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Dowker space

A Dowker space is a topological space which is normal but not countably paracompact.

C.H. Dowker had characterised these space in 1951 as those normal spaces for which the product with the unit interval is not normal, and asked whether any such space existed. M.E. Rudin constructed an example in 1971, and Zoltán Balogh gave the first ZFC construction of a small (cardinality continuum) example.

Erdős–Fuchs theorem

In mathematics, in the area of combinatorial number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of two elements of a given set, stating that the average order of this number cannot be close to being a linear function.

The theorem is named after Paul Erdős and Wolfgang Heinrich Johannes Fuchs.

Statement

Let A be a subset of the natural numbers and r(n) denote the number of ways that a natural number n can be expressed as the sum of two elements of A (taking order into account). We consider the average

\[R(n) = (r(1)+r(2)+\cdots+r(n) ) / n . \]

The theorem states that

\[R(n) = C + O\left(n^{-3/4-\epsilon}\right) \]

cannot hold unless C=0.

References

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Genus field

In algebraic number theory, the genus field $G$ of a number field $K$ is the maximal abelian extension of $K$ which is obtained by composing an absolutely abelian field with $K$ and which is unramified at all finite primes of $K$. The genus number of $K$ is the degree $[G:K]$ and the genus group is the Galois group of $G$ over $K$.

If $K$ is itself absolutely abelian, the genus field may be described as the maximal absolutely abelian extension of $K$ unramified at all finite primes: this definition was used by Leopoldt and Hasse.

If $K = \mathbb{Q}(\sqrt m)$ ($m$ squarefree) is a quadratic field of discriminant $D$, the genus field of $K$ is a composite of quadratic fields. Let $p_i$ run over the prime factors of $D$. For each such prime $p$, define $p^*$ as follows:

$$p^* = \pm p \equiv 1 \pmod 4 \text{ if } p \text{ is odd} ; $$ $$2^* = -4, 8, -8 \text{ according as } m \equiv 3 \pmod 4, 2 \pmod 8, -2 \pmod 8 . $$

Then the genus field is the composite $K(\sqrt{p^*_i})$.

See also

• Hilbert class field

References

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Group isomorphism problem

In abstract algebra, the group isomorphism problem is the decision problem of determining whether two group presentations present isomorphic groups.

The isomorphism problem was identified by Max Dehn in 1911 as one of three fundamental decision problems in group theory; the other two being the word problem and the conjugacy problem.

References

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Hall algebra

The Hall polynomials in mathematics were developed by Philip Hall in the 1950s in the study of group representations.

A finite abelian p-group M is a direct sum of cyclic p-power components \(C_{p^\lambda_i}\) where \(\lambda=(\lambda_1,\lambda_2,\ldots)\) is a partition of \(n\) called the type of M. Let \(g^\lambda_{\mu,\nu}(p)\) be the number of subgroups N of M such that N has type \(\nu\) and the quotient M/N has type \(\mu\). Hall showed that the functions g are polynomial functions of p with integer coefficients: these are the Hall polynomials.

Hall next constructs an algebra \(H(p)\) with symbols \(u_\lambda\) a generators and multiplication given by the \(g^\lambda_{\mu,\nu}\) as structure constants

\[ u_\mu u_\nu = \sum_\lambda g^\lambda_{\mu,\nu} u_\lambda \]

which is freely generated by the \(u_{\mathbf1_n}\) corresponding to the elementary p-groups. The map from \(H(p)\) to the algebra of symmetric functions \(e_n\) given by \(u_{\mathbf 1_n} \mapsto p^{-n(n-1)}e_n\) is a homomorphism and its image may be interpreted as the Hall-Littlewood polynomial functions. The theory of Schur functions is thus closely connected with the theory of Hall polynomials.

References

• I.G. Macdonald, Symmetric functions and Hall polynomials, (Oxford University Press, 1979) ISBN 0-19-853530-9

Hall–Littlewood polynomials

In mathematics, the Hall–Littlewood polynomials encode combinatorial data relating to the representations of the general linear group. They are named for Philip Hall and Dudley E. Littlewood.

See also

• Littlewood polynomial

References

• I.G. Macdonald; Symmetric Functions and Hall Polynomials, (1979), pp. 101-104, Oxford University Press ISBN: 0-19-853530-9
• D.E. Littlewood; On certain symmetric functions, Proc. London Math. Soc., 43 (1961), pp. 485–498, DOI: 10.1112/plms/s3-11.1.485

Hutchinson operator

In mathematics, in the study of fractals, a Hutchinson operator is a collection of functions on an underlying space E. The iteration on these functions gives rise to an iterated function system, for which the fixed set is self-similar.

Definition

Formally, let f_i be a finite set of N functions from a set X to itself. We may regard this as defining an operator H on the power set P X as

\[H : A \mapsto \bigcup_{i=1}^N f_i[A],\,\]

where A is any subset of X.

A key question in the theory is to describe the fixed sets of the operator H. One way of constructing such a fixed set is to start with an initial point or set S₀ and iterate the actions of the f_i, taking S_n+1 to be the union of the images of S_n under the operator H; then taking S to be the union of the S_n, that is,

\[S_{n+1} = \bigcup_{i=1}^N f_i[S_n] \]

and

\[S = \bigcup_{n=0}^\infty S_n . \]

Properties

Hutchinson (1981) considered the case when the f_i are contraction mappings on a Euclidean space X = R^d. He showed that such a system of functions has a unique compact (closed and bounded) fixed set S.

The collection of functions \(f_i\) together with composition form a monoid. With N functions, then one may visualize the monoid as a full N-ary tree or a Cayley tree.

References

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Manin obstruction

In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction is attached to a geometric object X which measures the failure of the Hasse principle for X: that is, if the value of the obstruction is non-trivial, then X may have points over all local fields but not over a global field.

For abelian varieties the Manin obstruction is just the Tate-Shafarevich group and fully accounts for the failure of the local-to-global principle. There are however examples, due to Skorobogatov, of varieties with trivial Manin obstruction which have points everywhere locally and yet no global points.

References

• Serge Lang. "Survey of Diophantine geometry". (Springer-Verlag, 1997) ISBN 3-540-61223-8. Zbl 0869.11051. pp.250–258.
• Alexei Skorobogatov (1999). "Beyond the Manin obstruction" (with Appendix A by S. Siksek: 4-descent). Inventiones Mathematicae 135 no.2 (1999) 399–424. DOI 10.1007/s002220050291. Zbl 0951.14013.
• Alexei Skorobogatov (2001). "Torsors and rational points". Cambridge Tracts in Mathematics 144 (Cambridge: Cambridge University Press, 2001). ISBN 0-521-80237-7. Zbl 0972.14015. pp.1–7,112.

Pinch point

A pinch point or cuspidal point is a type of singular point on an algebraic surface. It is one of the three types of ordinary singularity of a surface.

The equation for the surface near a pinch point may be put in the form

\[ f(u,v,w) = u^2 - vw^2 + [4] \, \]

where [4] denotes terms of degree 4 or more.

References

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Residual property

In the mathematical field of group theory, a group is residually X (where X is some property of groups) if it "can be recovered from groups with property X".

Formally, a group G is residually X if for every non-trivial element g there is a homomorphism h from G to a group with property X such that \(h(g)\neq e\).

More categorically, a group is residually X if it embeds into its pro-X completion (see profinite group, pro-p group), that is, the inverse limit of \(\phi\colon G \to H\) where H is a group with property X.

Examples

Important examples include:

• Residually finite
• Residually nilpotent
• Residually solvable
• Residually free

References

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