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Rings of constants of the form k[f]
... Theorem 2.2 (Zaks). If R is a Dedekind subring of k[X] containing k, then there exists a polynomial f ∈ k[X] such that R = k[f ]. Consider now the following family M of k-subalgebras of k[X]: M = {k[h]; h ∈ k[X] r k} . If k[h1 ] ( k[h2 ], for some h1 , h2 ∈ k[X] r k, then deg h2 < deg h1 and hence, ...
... Theorem 2.2 (Zaks). If R is a Dedekind subring of k[X] containing k, then there exists a polynomial f ∈ k[X] such that R = k[f ]. Consider now the following family M of k-subalgebras of k[X]: M = {k[h]; h ∈ k[X] r k} . If k[h1 ] ( k[h2 ], for some h1 , h2 ∈ k[X] r k, then deg h2 < deg h1 and hence, ...
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... The concept of reduction, also called multivariate division or normal form computation, is central to Gröbner basis theory. It is a multivariate generalization of the Euclidean division of univariate polynomials. In this section we suppose a fixed monomial ordering, which will not be defined explici ...
... The concept of reduction, also called multivariate division or normal form computation, is central to Gröbner basis theory. It is a multivariate generalization of the Euclidean division of univariate polynomials. In this section we suppose a fixed monomial ordering, which will not be defined explici ...
Unit 3B Notes: Graphs of Polynomial Functions
... Example 2 – CALC: Solve x4 – 3x3 + 5x2 – 27x – 36 = 0 by finding all roots. Step 1 Use the Rational Root Theorem to identify the possible rational roots. Step 2 Graph y = x4 – 3x3 + 5x2 – 27x – 36 to find the real roots. Step 3 Factor the polynomial using synthetic division. ...
... Example 2 – CALC: Solve x4 – 3x3 + 5x2 – 27x – 36 = 0 by finding all roots. Step 1 Use the Rational Root Theorem to identify the possible rational roots. Step 2 Graph y = x4 – 3x3 + 5x2 – 27x – 36 to find the real roots. Step 3 Factor the polynomial using synthetic division. ...
Finite fields / Galois Fields
... Let GF(q) be a (any) field of characteristic p, then it contains a prime-order subfield GF(p) = Z p = {0,1, 2 (1) ,3 (1) ,…, ( p − 1)(1)} . Proof. The set Z p = {0,1, 2 (1) ,3 (1) ,…, ( p − 1)(1)} contains p distinct elements because p is the characteristic of GF(q) and 0 have to be the first elemen ...
... Let GF(q) be a (any) field of characteristic p, then it contains a prime-order subfield GF(p) = Z p = {0,1, 2 (1) ,3 (1) ,…, ( p − 1)(1)} . Proof. The set Z p = {0,1, 2 (1) ,3 (1) ,…, ( p − 1)(1)} contains p distinct elements because p is the characteristic of GF(q) and 0 have to be the first elemen ...
Section X.56. Insolvability of the Quintic
... that K is a field which is an extension of F by radicals that includes all of the zeros of f (x). That is, we assume that f (x) is solvable by radicals and so the zeros are contained in an extension by radicals of F , say K. Then it must be that G(E/F ) is solvable. So if G(E/F ) is not a solvable g ...
... that K is a field which is an extension of F by radicals that includes all of the zeros of f (x). That is, we assume that f (x) is solvable by radicals and so the zeros are contained in an extension by radicals of F , say K. Then it must be that G(E/F ) is solvable. So if G(E/F ) is not a solvable g ...
ON DENSITY OF PRIMITIVE ELEMENTS FOR FIELD EXTENSIONS
... generates that field over the ground field. This is a fundamental theorem in algebra which is called the primitive element theorem in many textbooks, see for example [2, 5, 6], and it is a useful tool in practical computation of commutative algebra [4]. The existence proofs found in the literature m ...
... generates that field over the ground field. This is a fundamental theorem in algebra which is called the primitive element theorem in many textbooks, see for example [2, 5, 6], and it is a useful tool in practical computation of commutative algebra [4]. The existence proofs found in the literature m ...
PDF - Cryptology ePrint Archive
... number of iterations of the ECM is 265,557. Note that the digits of L computed from (10) number 25,234,114,168 for this B1 . 2.4.2 Results of Cheng Cheng’s method [2] is based on the ECM. Let N be an integer and p a prime factor of N such that p = (DV 2 + 1)/4, V ∈ Z, D ∈ {3, 11, 19, 43, 67, 163}. L ...
... number of iterations of the ECM is 265,557. Note that the digits of L computed from (10) number 25,234,114,168 for this B1 . 2.4.2 Results of Cheng Cheng’s method [2] is based on the ECM. Let N be an integer and p a prime factor of N such that p = (DV 2 + 1)/4, V ∈ Z, D ∈ {3, 11, 19, 43, 67, 163}. L ...
Finite-dimensional representations of difference
... Notation 1.1. Throughout this paper N is an arbitrary positive integer (unless otherwise explicitly indicated); N -vectors are denoted by underlined (Latin or Greek) letters, so that, for instance, the N vector v has the N components vn ; likewise (N × N )-matrices are denoted by twice-underlined (L ...
... Notation 1.1. Throughout this paper N is an arbitrary positive integer (unless otherwise explicitly indicated); N -vectors are denoted by underlined (Latin or Greek) letters, so that, for instance, the N vector v has the N components vn ; likewise (N × N )-matrices are denoted by twice-underlined (L ...
