Slides of the talk
... A NON - SOLVABLE POLYNOMIAL Consider the polynomial f (X ) = X 5 − 2X − 2, irreducible in Q[X ]. Its discriminant is 41808 = 24 · 3 · 13 · 67. ...
... A NON - SOLVABLE POLYNOMIAL Consider the polynomial f (X ) = X 5 − 2X − 2, irreducible in Q[X ]. Its discriminant is 41808 = 24 · 3 · 13 · 67. ...
FREE-BY-FREE GROUPS OVER POLYNOMIALLY GROWING
... CAT(0) if the metric induced by the Euclidean metric on the cubes turns it into a CAT(0) metric space (see [4]). In order to get the above statement, we prove the existence of a “space with walls” structure as introduced by Haglund and Paulin [12]. A theorem of Chatterji-Niblo [6] or Nica [15] (for ...
... CAT(0) if the metric induced by the Euclidean metric on the cubes turns it into a CAT(0) metric space (see [4]). In order to get the above statement, we prove the existence of a “space with walls” structure as introduced by Haglund and Paulin [12]. A theorem of Chatterji-Niblo [6] or Nica [15] (for ...
Ring Theory (Math 113), Summer 2014 - Math Berkeley
... that we cannot always divide, since 1/2 is no longer an integer. 2. Similarly, the familiar number systems Q, R, and C are all rings1 . 3. 2Z: the even integers ... , −4, −2, 0, 2, 4, .... 4. Z[x]: this is the set of polynomials whose coefficients are integers. It is an “extension” of Z in the sense ...
... that we cannot always divide, since 1/2 is no longer an integer. 2. Similarly, the familiar number systems Q, R, and C are all rings1 . 3. 2Z: the even integers ... , −4, −2, 0, 2, 4, .... 4. Z[x]: this is the set of polynomials whose coefficients are integers. It is an “extension” of Z in the sense ...
Universiteit Leiden Super-multiplicativity of ideal norms in number
... 4. every non-zero ideal is a power of m. A noetherian local domain of dimension one satisfying one of these conditions is called discrete valuation ring, briefly DVR. The proof can be found in [AM69, 9.2, pag. 94]. Then we have a criterion for invertibility for prime ideals of a number ring. Theorem ...
... 4. every non-zero ideal is a power of m. A noetherian local domain of dimension one satisfying one of these conditions is called discrete valuation ring, briefly DVR. The proof can be found in [AM69, 9.2, pag. 94]. Then we have a criterion for invertibility for prime ideals of a number ring. Theorem ...
Frobenius algebras and monoidal categories
... [Law1] F.W. Lawvere, Ordinal sums and equational doctrines, in: "Seminar on Triples and Categorical Homology Theory", Lecture Notes in Math. 80 (1969) 141–155. [Mü1] Michael Müger, From subfactors to categories and topology. I: Frobenius algebras in and Morita equivalence of tensor categories, J. Pu ...
... [Law1] F.W. Lawvere, Ordinal sums and equational doctrines, in: "Seminar on Triples and Categorical Homology Theory", Lecture Notes in Math. 80 (1969) 141–155. [Mü1] Michael Müger, From subfactors to categories and topology. I: Frobenius algebras in and Morita equivalence of tensor categories, J. Pu ...
dmodules ja
... object and we hope to study -modules on X by using these methods. In particular, effective algorithms have been developed for -modules on affine space; for example, see the work of Oaku [12], Walther [16], Saito et al. [15], and Oaku and Takayama [13]. It would be interesting to use our results to ...
... object and we hope to study -modules on X by using these methods. In particular, effective algorithms have been developed for -modules on affine space; for example, see the work of Oaku [12], Walther [16], Saito et al. [15], and Oaku and Takayama [13]. It would be interesting to use our results to ...
A UNIFORM OPEN IMAGE THEOREM FOR l
... The main technical tool we resort to is that, for any integer γ ≥ 1 there exists an integer ν = ν(γ) ≥ 1 such that, given any projective system · · · → Yn+1 → Yn → · · · → Y0 of curves with the same gonality γ and with Yn+1 → Yn a Galois cover of degree > 1, one can construct a projective system of ...
... The main technical tool we resort to is that, for any integer γ ≥ 1 there exists an integer ν = ν(γ) ≥ 1 such that, given any projective system · · · → Yn+1 → Yn → · · · → Y0 of curves with the same gonality γ and with Yn+1 → Yn a Galois cover of degree > 1, one can construct a projective system of ...
Undergraduate algebra
... the rectangle have the same number of symmetries, but they are clearly symmetric in different ways. How can one capture this difference? Given two symmetries of some shape, we may transform the shape by the first one, and then apply the second one to the result. The operation obtained in this way is ...
... the rectangle have the same number of symmetries, but they are clearly symmetric in different ways. How can one capture this difference? Given two symmetries of some shape, we may transform the shape by the first one, and then apply the second one to the result. The operation obtained in this way is ...
From prime numbers to irreducible multivariate polynomials
... i=0 i h , with a0 , a1 , . . . , an ∈ K[X]. Then the polynomial Pn i i=0 ai (X)Y is irreducible over K(X). Corollary 3.4. Let K be a field, f, g, h ∈ K[X], f irreducible over K, s g 6= 0, deg g < deg h, and assume that for an integer s ≥ 2 the polynomial Pn f ·g s is expressed “in base” h via the P ...
... i=0 i h , with a0 , a1 , . . . , an ∈ K[X]. Then the polynomial Pn i i=0 ai (X)Y is irreducible over K(X). Corollary 3.4. Let K be a field, f, g, h ∈ K[X], f irreducible over K, s g 6= 0, deg g < deg h, and assume that for an integer s ≥ 2 the polynomial Pn f ·g s is expressed “in base” h via the P ...
RULED SURFACES WITH NON-TRIVIAL SURJECTIVE
... fibration h : X → C onto a non-singular curve C. The fibers of π dominate C. Hence C P1 . Let D be a general fiber of h. Then D2 = 0 and π(D) = B. (2) =⇒ (3). If there is a section C0 of π with C02 < 0, then any other irreducible curve C with π(C) = B is linearly equivalent to aC0 + π ∗E for some a > ...
... fibration h : X → C onto a non-singular curve C. The fibers of π dominate C. Hence C P1 . Let D be a general fiber of h. Then D2 = 0 and π(D) = B. (2) =⇒ (3). If there is a section C0 of π with C02 < 0, then any other irreducible curve C with π(C) = B is linearly equivalent to aC0 + π ∗E for some a > ...
Open Mapping Theorem for Topological Groups
... (ii) Homomorphic images of Q are either singleton or infinite. (iii) A finite group has no divisible subgroups other than the singleton one. Proof. (i) Using divisibility, recursively define elements g1 = g, g2 , . . . such that gnn = gn−1 , n = 2, 3, . . .. Every rational number q ∈ Q can be writte ...
... (ii) Homomorphic images of Q are either singleton or infinite. (iii) A finite group has no divisible subgroups other than the singleton one. Proof. (i) Using divisibility, recursively define elements g1 = g, g2 , . . . such that gnn = gn−1 , n = 2, 3, . . .. Every rational number q ∈ Q can be writte ...
HW 2
... (a) Let fg be the inner automorphism given by fg (a) = gag −1 and let ϕ be any automorphism of g. Then for a ∈ G, (ϕ ◦ fg ◦ ϕ−1 )(a)ϕ ◦ fg (ϕ−1 (a)) = ϕ(gϕ−1 ag −1 ) = ϕ(g)ϕ(ϕ−1 (a))ϕ(g −1 ) = ϕ(g)aϕ(g)−1 , where ϕ(ϕ−1 (a)) = a because ϕ is an automorphism, hence bijective. Therefore ϕ ◦ fg ϕ−1 = fϕ ...
... (a) Let fg be the inner automorphism given by fg (a) = gag −1 and let ϕ be any automorphism of g. Then for a ∈ G, (ϕ ◦ fg ◦ ϕ−1 )(a)ϕ ◦ fg (ϕ−1 (a)) = ϕ(gϕ−1 ag −1 ) = ϕ(g)ϕ(ϕ−1 (a))ϕ(g −1 ) = ϕ(g)aϕ(g)−1 , where ϕ(ϕ−1 (a)) = a because ϕ is an automorphism, hence bijective. Therefore ϕ ◦ fg ϕ−1 = fϕ ...
Group Theory G13GTH
... with all elements k ∈ K, which is not hard to check on the six elements directly. Now let g ∈ G be any element. If g fixes the triangle then g ∈ K and g = 1g ∈ HK. Otherwise hg will fix the triangle and hence g = h(hg) ∈ HK. Since H ∩ K = {1}, we have shown that G = H × K. Proposition 1.7. Let {1} 6 ...
... with all elements k ∈ K, which is not hard to check on the six elements directly. Now let g ∈ G be any element. If g fixes the triangle then g ∈ K and g = 1g ∈ HK. Otherwise hg will fix the triangle and hence g = h(hg) ∈ HK. Since H ∩ K = {1}, we have shown that G = H × K. Proposition 1.7. Let {1} 6 ...
Closed sets and the Zariski topology
... so all cγ = 0 except for the cγ where xαi +γ = xβ , and for these cγ = 1. n Hence there is some γ ∈ Z+ such that β = α + γ, and the result follows. Exercise 3.1 (10 pushups). Consider Z2+ , and let A = {(1, 2), (2, 0), (3, 1), (2, 2)}. Draw a diagram of A and use the picture to find a minimal set ...
... so all cγ = 0 except for the cγ where xαi +γ = xβ , and for these cγ = 1. n Hence there is some γ ∈ Z+ such that β = α + γ, and the result follows. Exercise 3.1 (10 pushups). Consider Z2+ , and let A = {(1, 2), (2, 0), (3, 1), (2, 2)}. Draw a diagram of A and use the picture to find a minimal set ...
Advanced Algebra - Stony Brook Mathematics
... to imitate the theory of “Lie algebras,” which began about 1880. A brief summary of some early theorems about Lie algebras will put matters in perspective. The term “algebra” in connection with a field F refers at least to an F vector space with a multiplication that is F bilinear. This chapter will ...
... to imitate the theory of “Lie algebras,” which began about 1880. A brief summary of some early theorems about Lie algebras will put matters in perspective. The term “algebra” in connection with a field F refers at least to an F vector space with a multiplication that is F bilinear. This chapter will ...
1 - Evan Chen
... As far as presentations, we have D2n = < r, s |rn = s2 = 1, rs = sr−1 > It’s common the relations are the orders of the generators. A presentation is not unique, however. This particular presentation is useful because each element can be written as rk s` for ` ∈ {0, 1} and k ∈ {0, 1, · · · , n−1}. P ...
... As far as presentations, we have D2n = < r, s |rn = s2 = 1, rs = sr−1 > It’s common the relations are the orders of the generators. A presentation is not unique, however. This particular presentation is useful because each element can be written as rk s` for ` ∈ {0, 1} and k ∈ {0, 1, · · · , n−1}. P ...
pdf-file. - Fakultät für Mathematik
... the paper which speaks of a “normed” multiplicative basis). The multiplicativity property (1) asserts that one deals with a combinatorially defined algebra; however in applications also the remaining properties turn out to be of great importance: after all, any group algebra has, for trivial reasons ...
... the paper which speaks of a “normed” multiplicative basis). The multiplicativity property (1) asserts that one deals with a combinatorially defined algebra; however in applications also the remaining properties turn out to be of great importance: after all, any group algebra has, for trivial reasons ...
on h1 of finite dimensional algebras
... algebra itself, H 0 (Λ, Λ) is the center Z(Λ) of Λ. The trivial case corresponds to an algebra Λ which is projective as a Λe -left module. This occurs only when Λ is semisimple (a product of matrix algebras over finite dimensional skew-fields) and the centers of the skew-fields are separable extensi ...
... algebra itself, H 0 (Λ, Λ) is the center Z(Λ) of Λ. The trivial case corresponds to an algebra Λ which is projective as a Λe -left module. This occurs only when Λ is semisimple (a product of matrix algebras over finite dimensional skew-fields) and the centers of the skew-fields are separable extensi ...
Classifying classes of structures in model theory
... Let us ignore the types of elements a ∈ M, as they are easy to understand; so assume a ∈ N \ M. Assume that M ⊆ N are algebraically closed fields. Then all elements b ∈ N \ M have the same type, so there is only one nontrivial type over M. If M is a dense linear order, there are always many nontrivi ...
... Let us ignore the types of elements a ∈ M, as they are easy to understand; so assume a ∈ N \ M. Assume that M ⊆ N are algebraically closed fields. Then all elements b ∈ N \ M have the same type, so there is only one nontrivial type over M. If M is a dense linear order, there are always many nontrivi ...
Introduction - SUST Repository
... multiplication . In this case we often use ∗ or a dot , to denote the operation and write a ∗ b as ab for brevity . We often denote the identity by e or 1,and the inverse of a in G as a-1 . . Note that in our group axioms above we don’t assume commutatively ( which means that if we have any x and y ...
... multiplication . In this case we often use ∗ or a dot , to denote the operation and write a ∗ b as ab for brevity . We often denote the identity by e or 1,and the inverse of a in G as a-1 . . Note that in our group axioms above we don’t assume commutatively ( which means that if we have any x and y ...
Ring Theory
... mechanisms by which the subject progresses. The definition of a ring consists of a list of technical properties, but the motivation for this definition is the ubiquity of objects having these properties, like the ones in Section 1.1. When making a definition like that of a ring (or group or vector s ...
... mechanisms by which the subject progresses. The definition of a ring consists of a list of technical properties, but the motivation for this definition is the ubiquity of objects having these properties, like the ones in Section 1.1. When making a definition like that of a ring (or group or vector s ...
The local structure of twisted covariance algebras
... irreducible representations. I t is known that when G is not type I, G^, with the Mackey Borel structure, is not standard, or even countably separated. This is generally interpreted to mean that the irreducible representations of such a group are not classifiable, and so the problem becomes to find ...
... irreducible representations. I t is known that when G is not type I, G^, with the Mackey Borel structure, is not standard, or even countably separated. This is generally interpreted to mean that the irreducible representations of such a group are not classifiable, and so the problem becomes to find ...
Lattices in Lie groups
... Groups. In Section 2, we will collect some general results on lattices in locally compact groups. The first main theorem is that the discrete subgroup SLn (Z) is a lattice in SLn (R). This will be proved in section 4. We then prove the Mahler criterion, which will enable us to prove the co-compactne ...
... Groups. In Section 2, we will collect some general results on lattices in locally compact groups. The first main theorem is that the discrete subgroup SLn (Z) is a lattice in SLn (R). This will be proved in section 4. We then prove the Mahler criterion, which will enable us to prove the co-compactne ...
Chern Character, Loop Spaces and Derived Algebraic Geometry
... Abstract In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of O-modules on schemes, as well as its quasi-coher ...
... Abstract In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of O-modules on schemes, as well as its quasi-coher ...
Contemporary Abstract Algebra (6th ed.) by Joseph Gallian
... Z17 . Since |Z17 | = 17 is prime, Ker φ can either be of size 1 or 17. By theorem 10.2, property 5, |Ker φ| 6= 1, so it must 17. Thus phi is the trivial map, that is it takes all elements of Z17 and maps them to the identity of G. 10.22 Suppose that φ is a homomorphism from a finite group G onto G a ...
... Z17 . Since |Z17 | = 17 is prime, Ker φ can either be of size 1 or 17. By theorem 10.2, property 5, |Ker φ| 6= 1, so it must 17. Thus phi is the trivial map, that is it takes all elements of Z17 and maps them to the identity of G. 10.22 Suppose that φ is a homomorphism from a finite group G onto G a ...