Algebra Qualifying Exam Notes
... Proposition 5: If G is a simple group and H is a proper subgroup of G of index n, then there is an injective homomorphisms φ : G → Sn . Proof: Let G act on the cosets of H by left multiplication. This action permutes the cosets of H. So it induces a homomorphism φ : G → S[G:H] ∼ = Sn . Note that φ i ...
... Proposition 5: If G is a simple group and H is a proper subgroup of G of index n, then there is an injective homomorphisms φ : G → Sn . Proof: Let G act on the cosets of H by left multiplication. This action permutes the cosets of H. So it induces a homomorphism φ : G → S[G:H] ∼ = Sn . Note that φ i ...
noncommutative polynomials nonnegative on a variety intersect a
... 1.3. Linear Pencils. A linear pencil is a symmetric polynomial L ∈ Rν×ν hx, x∗ i, for some ν ∈ N, with deg(L) ≤ 1. Every ν × ν linear pencil can be expressed as L = A0 + A1 ⊗ x1 + · · · + Ag ⊗ xg + A∗1 ⊗ x∗1 + · · · + A∗g ⊗ x∗g , where each Ai ∈ Rν×ν and A0 is symmetric. A linear pencil is monic if ...
... 1.3. Linear Pencils. A linear pencil is a symmetric polynomial L ∈ Rν×ν hx, x∗ i, for some ν ∈ N, with deg(L) ≤ 1. Every ν × ν linear pencil can be expressed as L = A0 + A1 ⊗ x1 + · · · + Ag ⊗ xg + A∗1 ⊗ x∗1 + · · · + A∗g ⊗ x∗g , where each Ai ∈ Rν×ν and A0 is symmetric. A linear pencil is monic if ...
Henry Cohn`s home page
... Theorem 2.6 (Unique Factorization in Z). Factorization of non-zero integers into primes is unique, up to rearrangement of the factors and change of their signs. Note that we consider 1 to have the empty prime factorization: it is the product of no primes. Proof. We first need to establish the existe ...
... Theorem 2.6 (Unique Factorization in Z). Factorization of non-zero integers into primes is unique, up to rearrangement of the factors and change of their signs. Note that we consider 1 to have the empty prime factorization: it is the product of no primes. Proof. We first need to establish the existe ...
Rational Polynomial Pell Equations - Mathematics
... Such an identity has the potential for applications to the problem of computing class numbers of real quadratic number fields. If, for example, we know that ...
... Such an identity has the potential for applications to the problem of computing class numbers of real quadratic number fields. If, for example, we know that ...
Lecture 8: Stream ciphers - LFSR sequences
... where T is the order or α. If α is a primitive element (its order is q L − 1), then obviously we will go trough all q L − 1 different states and the sequence will have period q L − 1. Such sequences are called m-sequences and they appear if and only if the polynomial π(x) is a primitive polynomial. ...
... where T is the order or α. If α is a primitive element (its order is q L − 1), then obviously we will go trough all q L − 1 different states and the sequence will have period q L − 1. Such sequences are called m-sequences and they appear if and only if the polynomial π(x) is a primitive polynomial. ...
Notes on Ring Theory
... Definition: Characteristic of a ring. A ring R with 1 has characteristic n if n is the first positive integer for which 1 + 1 + · · · n terms = 0. In case, such an n does not exist, the characteristic is said to be 0. A simpler description of the characteristic is as follows. Define a homomorphism φ ...
... Definition: Characteristic of a ring. A ring R with 1 has characteristic n if n is the first positive integer for which 1 + 1 + · · · n terms = 0. In case, such an n does not exist, the characteristic is said to be 0. A simpler description of the characteristic is as follows. Define a homomorphism φ ...
Lecture 2 - Stony Brook Mathematics
... Consider the set A of all pairs of natural numbers (a, b) for which a, b > 0 and a2 = 2b 2 . If there is a rational solution to x 2 = 2, then A is non-empty, hence, by the well-ordering principle, there is a pair (a0 , b0 ) which minimizes a0 + b0 . Then a0 is even, a0 = 2a1 . It follows that 4a12 = ...
... Consider the set A of all pairs of natural numbers (a, b) for which a, b > 0 and a2 = 2b 2 . If there is a rational solution to x 2 = 2, then A is non-empty, hence, by the well-ordering principle, there is a pair (a0 , b0 ) which minimizes a0 + b0 . Then a0 is even, a0 = 2a1 . It follows that 4a12 = ...
Ring Theory (Math 113), Summer 2014 - Math Berkeley
... and add, giving rise to x 2 , x 3 , etc., as well as 2x, 3x, etc. Adding up various combinations of these gives all the possible integer polynomials. 5. Z[x, y , z]: polynomials in three variables with integer coefficients. This is an extension of the previous ring, too. In fact you can continue add ...
... and add, giving rise to x 2 , x 3 , etc., as well as 2x, 3x, etc. Adding up various combinations of these gives all the possible integer polynomials. 5. Z[x, y , z]: polynomials in three variables with integer coefficients. This is an extension of the previous ring, too. In fact you can continue add ...
Number Theory - Fredonia.edu
... One of the fundamental concepts included in any introduction to number theory is that of factoring integers. In particular, given an integer n > 1, we are interested in expressing n as a product of primes. For example, if n = 132, then n = 2 · 2 · 3 · 11. Is it always possible to do this? Can it, fo ...
... One of the fundamental concepts included in any introduction to number theory is that of factoring integers. In particular, given an integer n > 1, we are interested in expressing n as a product of primes. For example, if n = 132, then n = 2 · 2 · 3 · 11. Is it always possible to do this? Can it, fo ...
KNOT SIGNATURE FUNCTIONS ARE INDEPENDENT 1
... Previously the only sets D for which it was known that {σω }ω∈D is linearly independent were certain discrete subsets of S, [7, 13]. Applications of this result include the demonstration that a number of results that hold in high-dimensional knot theory fail in dimension 3. This is briefly summarize ...
... Previously the only sets D for which it was known that {σω }ω∈D is linearly independent were certain discrete subsets of S, [7, 13]. Applications of this result include the demonstration that a number of results that hold in high-dimensional knot theory fail in dimension 3. This is briefly summarize ...
4. Rings 4.1. Basic properties. Definition 4.1. A ring is a set R with
... Much of the basic material on groups just carries over to rings (or other algebraic structures) in a very straightforward way. We already defined subrings. If R, R0 are rings, then a map ϕ : R → R0 is called a homomorphism if ϕ(a + b) = ϕ(a) + ϕ(b), ϕ(ab) = ϕ(a)ϕ(b), ϕ(1) = 10 . Equivalently, we ask ...
... Much of the basic material on groups just carries over to rings (or other algebraic structures) in a very straightforward way. We already defined subrings. If R, R0 are rings, then a map ϕ : R → R0 is called a homomorphism if ϕ(a + b) = ϕ(a) + ϕ(b), ϕ(ab) = ϕ(a)ϕ(b), ϕ(1) = 10 . Equivalently, we ask ...