Geometric Constructions from an Algebraic Perspective
... Theorem 24. Let α be a real number. Then α is constructible if and only if α belongs to the top of some square root tower over Q. Proof. (⇐)Let C be the set of constructible real numbers. C is an extension field of Q. C is a subfield of R because we have shown earlier that the constructible set C is ...
... Theorem 24. Let α be a real number. Then α is constructible if and only if α belongs to the top of some square root tower over Q. Proof. (⇐)Let C be the set of constructible real numbers. C is an extension field of Q. C is a subfield of R because we have shown earlier that the constructible set C is ...
LINEAR INDEPENDENCE OF LOGARITHMS OF - IMJ-PRG
... a1 · · · abmm − 1 ≥ A−nB where A = max{2, a1 , . . . , an } and B = max{2, |b1 |, . . . , |bn |}; this is a kind of Liouville estimate: the absolute value of a non-zero rational number is at least the inverse of a denominator. This estimate is sharp in terms of A and n, but not in terms of B; ther ...
... a1 · · · abmm − 1 ≥ A−nB where A = max{2, a1 , . . . , an } and B = max{2, |b1 |, . . . , |bn |}; this is a kind of Liouville estimate: the absolute value of a non-zero rational number is at least the inverse of a denominator. This estimate is sharp in terms of A and n, but not in terms of B; ther ...
Elements of Modern Algebra
... A minimal amount of mathematical maturity is assumed in the text; a major goal is to develop mathematical maturity. The material is presented in a theorem-proof format, with definitions and major results easily located thanks to a user-friendly format. The treatment is rigorous and self-contained, i ...
... A minimal amount of mathematical maturity is assumed in the text; a major goal is to develop mathematical maturity. The material is presented in a theorem-proof format, with definitions and major results easily located thanks to a user-friendly format. The treatment is rigorous and self-contained, i ...
a) - BrainMass
... a) Define the greatest common divisor of two integers. Let a and b be two integers . Then the greatest common divisor of a and b is defined as the largest positive integer k such that a and b are both divisible by k. Denoted by GCD(a,b). For instance, the gcd(2,4)=2 ...
... a) Define the greatest common divisor of two integers. Let a and b be two integers . Then the greatest common divisor of a and b is defined as the largest positive integer k such that a and b are both divisible by k. Denoted by GCD(a,b). For instance, the gcd(2,4)=2 ...
Full text
... for a natural algebraic and geometric setting for their analysis. In this way many known results are unified and simplified and new results are obtained. Some of the results extend to Fibonacci representations of higher order, but we do not present these because we have been unable to extend the the ...
... for a natural algebraic and geometric setting for their analysis. In this way many known results are unified and simplified and new results are obtained. Some of the results extend to Fibonacci representations of higher order, but we do not present these because we have been unable to extend the the ...
4 Number Theory 1 4.1 Divisors
... Note: This definition satisfies gcd(0,1) = 1. The lowest common multiple lcm(a,b) is defined as follows: lcm(a,b) = min(m > 0 s.t. a|m and b|m) (for a 6= 0 and b 6= 0). a and b are coprimes (or relatively prime) iff gcd(a,b) = 1. Prime Numbers An integer p ≥ 2 is called prime if it is divisible only ...
... Note: This definition satisfies gcd(0,1) = 1. The lowest common multiple lcm(a,b) is defined as follows: lcm(a,b) = min(m > 0 s.t. a|m and b|m) (for a 6= 0 and b 6= 0). a and b are coprimes (or relatively prime) iff gcd(a,b) = 1. Prime Numbers An integer p ≥ 2 is called prime if it is divisible only ...
Factorization
In mathematics, factorization (also factorisation in some forms of British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms.Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. This situation is generalized by factorization systems.