Narrow Results

Let $f(x)=x^8+a{x}^4+b$ be an irreducible polynomial with rational coefficients and Galois group $\mathrm{\text{Gal}}(f)$. We extend previous results to give an elementary classification of $\mathrm{\text{Gal}}(f)$, identified as a transitive subgroup of $S_8$ up to conjugacy. We show $\mathrm{\text{Gal}}(f)$ is one of 12 possibilities and can be determined by considering the squareness of at most 11 rational numbers; each number is an expression involving $a$ and $b$. We give several applications of our results.

We prove that the images of irreducible germs of plane curves by a germ of analytic morphism φ have a certain contact either with branches of the discriminant of φ or with certain infinitesimal structures (shadows) that arise from the branches of the Jacobian of φ that are mapped to a point (and therefore give rise to no branch of the discriminant).

We study the rigidity of three-dimensional representations of braid groups associated with finite primitive irreducible complex reflection groups in $\mathrm{\text{GL}}(3,ℂ)$. In many cases, we show the rigidity. For rigid representations, we give explicit forms of the representations, which turns out to be the monodromy representations of uniformization equations of Saito–Kato–Sekiguchi [Uniformization systems of equations with singularities along the discriminant sets of complex reflection groups of rank three, Kyushu J. Math.68 (2014) 181–221; On the uniformization of complements of discriminant loci, RIMS Kokyuroku287 (1977) 117–137]. Invariant Hermitian forms are also studied.

In [11], Hausel introduced a commutative algebra — the multiplicity algebra — associated to a fixed point of the $C^{\ast }$-action on the Higgs bundle moduli space. Here we describe this algebra for a fixed point consisting of a very stable rank 2 vector bundle and zero Higgs field for a curve of low genus. Geometrically, the relations in the algebra are described by a family of quadrics and we focus on the discriminant of this family, providing a new viewpoint on the moduli space of stable bundles. The discriminant in our examples demonstrates that as the bundle varies, we obtain a continuous variation in the isomorphism class of the algebra.

In this paper we describe complexes whose homologies are naturally isomorphic to the first term of the Vassiliev spectral sequence computing (co)homology of the spaces of long knots in ℝ^d, d ≥ 3. The first term of the Vassiliev spectral sequence is concentrated in some angle of the second quadrant. In homological case the lower line of this term is the bialgebra of chord diagrams (or its superanalog if d is even). We prove in this paper that the groups of the upper line are all trivial. In the same bigradings we compute the homology groups of the complex spanned only by strata of immersions in the discriminant (maps having only self-intersections). We interpret the obtained groups as subgroups of the (co)homology groups of the double loop space of a (d - 1)-dimensional sphere. In homological case the last complex is the normalized Hochschild complex of the Poisson or Gerstenhaber (depending on parity of d) algebras operad. The upper line bigradings are spanned by the operad of Lie algebras. To describe the cycles in these bigradings, we introduce new homological operations on Hochschild complexes. We show in future work that these operations are in fact the Dyer–Lashof–Cohen operations induced by the action of the singular chains operad of little squares on Hochschild complexes.

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point p of the cylinder is called coherent if all three branches intersect at p pairwise with the same intersection index. A triple unknotting of a classical knot K is a homotopy which connects K with the trivial knot and which has as singularities only coherent triple points.

We give a new formula for the first Vassiliev invariant v₂(K) by using triple unknottings. As a corollary we obtain a very simple proof of the fact that passing a coherent triple point always changes the knot type. As another corollary we show that there are triple unknottings which are not homotopic as triple unknottings even if we allow more complicated singularities to appear in the homotopy of the triple homotopy.

The construction of integer linking numbers of closed curves in a three-dimensional manifold usually appeals to the orientation of this manifold. We discuss how to avoid it constructing similar homotopy invariants of links in non-orientable manifolds.

A formula for computing the discriminant of any number field K ⊂ ℚ(ζ_2^r), with r ≥ 3, is derived. The formula consists of two expressions, depending on whether K is cyclotomic or not. However, both expressions depend solely on m, the degree of K over ℚ, and they are derived from the Conductor-Discriminant Formula for Abelian extensions of ℚ.

Let $f(x)=x^8+a{x}^4+b$ be an irreducible polynomial with rational coefficients, $K/Q$ the number field defined by $f$, and $G$ the Galois group of $f$. Let $g(x)=x^4+a{x}^2+b$, and let $G_4$ be the Galois group of $g$. We investigate the extent to which knowledge of the conjugacy class of $G_4$ in $S_4$ determines the conjugacy class of $G$ in $S_8$. We show that, in general, knowledge of $G_4$ does not automatically determine $G$, except when $G_4$ is isomorphic to $C_4$ (the cyclic group of order 4). In this case, we show $G$ is isomorphic to a non-split extension of $D_4$ (the dihedral group of order 8) by $C_4$. We also show that $G$ is completely determined when $G_4$ is isomorphic to $D_4$ and $4b-a^2$ is a perfect square. In this case, $G\simeq C_4\wr C_2\simeq (C_4\times C_4)⋊C_2$.

Let $K=\mathbb{Q}(𝜃)$ be an algebraic number field with $𝜃$ a root of an irreducible trinomial $f(x)=x^5+ax+b$ belonging to $\mathbb{Z}[x]$. In this paper, we compute the highest power of each prime $p$ dividing the discriminant of $K$ in terms of powers of $p$ dividing $a,b$ and the discriminant of $f(x)$ besides explicitly constructing a $p$-integral basis of $K$. These $p$-integral bases lead to the construction of an integral basis of $K$ which is illustrated with examples.

This paper concerns a classical subject regarding the structural properties of abelian number fields. The primitive elements, integral bases, conductors and discriminants are important for studying abelian number fields, and this paper emphasizes that they are closely related to the associated Dirichlet characters. Then by evaluating the Gauss sums, we can give explicit forms of abelian fields of small degree.

Similarity is an important and widely used concept in many applications such as Document Summarisation, Question Answering, Information Retrieval, Document Clustering and Categorisation. This paper presents a comparison of various similarity measures in comparing the content of text documents. We have attempted to find the best measure suited for finding the document similarity for newspaper reports.

In this paper we prove an analogue of Mertens' theorem for primes of each of the forms a²+27b² and 4a²+2ab+7b² and then use this result to determine an asymptotic formula for the number of positive integers n ≤ x which are discriminants of cyclic cubic fields with each such field having field index 2.

For a finite abelian extension K/ℚ, the conductor-discriminant formula establishes that the absolute value of the discriminant of K is equal to the product of the conductors of the elements of the group of Dirichlet characters associated to K. The simplest proof uses the functional equation for the Dedekind zeta function of K and its expression as the product of the L-series attached to the various Dirichlet characters associated to K. In this paper, we present an elementary proof of this formula considering first K contained in a cyclotomic number field of pⁿ-roots of unity, where p is a prime number, and in the general case, using the ramification index of p given by the group of Dirichlet characters.

Let G = C_ℓ × C_ℓ denote the product of two cyclic groups of prime order ℓ, and let k be an algebraic number field. Let N(k, G, m) denote the number of abelian extensions K of k with Galois group G(K/k) isomorphic to G, and the relative discriminant 𝒟(K/k) of norm equal to m. In this paper, we derive an asymptotic formula for ∑_m≤XN(k, G; m). This extends the result previously obtained by Datskovsky and Mammo.

We present the first explicitly known polynomials in Z[x] with nonsolvable Galois group and field discriminant of the form ±p^A for p ≤ 7 a prime. Our main polynomial has degree 25, Galois group of the form PSL₂(5)⁵.10, and field discriminant 5⁶⁹. A closely related polynomial has degree 120, Galois group of the form SL₂(5)⁵.20, and field discriminant 5³¹¹. We completely describe 5-adic behavior, finding in particular that the root discriminant of both splitting fields is 125 · 5^-1/12500 ≈ 124.984 and the class number of the latter field is divisible by 5⁴.

Let p denote an odd prime. For all p-admissible conductors c over a quadratic number field , p-ring spaces V_p(c) modulo c are introduced by defining a morphism ψ : f ↦ V_p(f) from the divisor lattice ℕ of positive integers to the lattice 𝒮 of subspaces of the direct product V_p of the p-elementary class group 𝒞/𝒞^p and unit group U/U^p of K. Their properties admit an exact count of all extension fields N over K, having the dihedral group of order 2p as absolute Galois group Gal(N | ℚ) and sharing a common discriminant d_N and conductor c over K. The number m_p(d, c) of these extensions is given by a formula in terms of positions of p-ring spaces in 𝒮, whose complexity increases with the dimension of the vector space V_p over the finite field 𝔽_p, called the modified p-class rank σ_p of K. Up to now, explicit multiplicity formulas for discriminants were known for quadratic fields with 0 ≤ σ_p ≤ 1 only. Here, the results are extended to σ_p = 2, underpinned by concrete numerical examples.

We tabulate polynomials in ℚ[t] with a given factorization partition, bad reduction entirely within a given set of primes, and satisfying auxiliary conditions associated to 0, 1, and ∞. We explain how these polynomials are of particular interest because of their role in the construction of nonsolvable number fields of arbitrarily large degree and bounded ramification.

For any square-free polynomial D over a finite field of characteristic at least 5, we present an algorithm for generating all cubic function fields of discriminant D. We also provide a count of all these fields according to their splitting at infinity. When D′ = D/(-3) has even degree and a leading coefficient that is a square, i.e. D′ is the discriminant of a real quadratic function field, this method makes use of the infrastructures of this field. This infrastructure method was first proposed by Shanks for cubic number fields in an unpublished manuscript from the late 1980s. While the mathematical ingredients of our construction are largely classical, our algorithm has the major computational advantage of finding very small minimal polynomials for the fields in question.

We prove a general theorem that evaluates the Legendre symbol under certain conditions on the integers A, B, m and the prime p. The evaluation is in terms of parameters appearing in a binary quadratic form representing p. The theorem has applications to quartic residuacity.