GROUPS, RINGS AND FIELDS
... To see, that (4.3) works, let d=gcd(a,b). Then, by the definition of gcd, d|a and d|b. For any positive integer b, a can be expressed in the form a = kb+r r mod b a mod b = r with k, r integers. Therefore, (a mod b) = a-kb for some integer k. But because d|b, it also divides kb. We also have d|a. ...
... To see, that (4.3) works, let d=gcd(a,b). Then, by the definition of gcd, d|a and d|b. For any positive integer b, a can be expressed in the form a = kb+r r mod b a mod b = r with k, r integers. Therefore, (a mod b) = a-kb for some integer k. But because d|b, it also divides kb. We also have d|a. ...
Class Notes - College of Engineering and Applied Science
... three positive integers, viz., 1, 2 and 3. In such cases, we need to exclude the smallest positive integers from consideration. A specific example that can be provide by induction is that n! > 2n for all n ≥ 4. That is, this result holds only for positive integers 4, 5, 6, · · ·. ...
... three positive integers, viz., 1, 2 and 3. In such cases, we need to exclude the smallest positive integers from consideration. A specific example that can be provide by induction is that n! > 2n for all n ≥ 4. That is, this result holds only for positive integers 4, 5, 6, · · ·. ...
Number Theory - Redbrick DCU
... these numbers requires us to modify the standard rules of arithmetic, basically so that the result of any operation is a number in the same range. On an actual 12 hour clock face p=12, and 9 O’Clock plus 4 hours gives us 1 O’Clock. What we have actually done is to calculate 9+4 -12. The addition rul ...
... these numbers requires us to modify the standard rules of arithmetic, basically so that the result of any operation is a number in the same range. On an actual 12 hour clock face p=12, and 9 O’Clock plus 4 hours gives us 1 O’Clock. What we have actually done is to calculate 9+4 -12. The addition rul ...
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 ...
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.