THE INTEGERS 1. Divisibility and Factorization Without discussing
... details than the previous one, but it points to a far-reaching ideas in its own right: if p = 0 then not only is the pth power of a product inevitably the product of the pth powers, (ab)p = ap bp , but also the pth power of a sum is the sum of the pth powers, (a + b)p = ap + bp . The fact that raisi ...
... details than the previous one, but it points to a far-reaching ideas in its own right: if p = 0 then not only is the pth power of a product inevitably the product of the pth powers, (ab)p = ap bp , but also the pth power of a sum is the sum of the pth powers, (a + b)p = ap + bp . The fact that raisi ...
Primes, Polygons, and Polynomials
... Gauss did not actually show the construction of the 17-gon. This was done a few years later. In 1832, a description of the construction of a 257-gon was published; the description took over 200 pages! It would have to be a big 257-gon, or it would look like a circle, and I can’t imagine how much acc ...
... Gauss did not actually show the construction of the 17-gon. This was done a few years later. In 1832, a description of the construction of a 257-gon was published; the description took over 200 pages! It would have to be a big 257-gon, or it would look like a circle, and I can’t imagine how much acc ...
(pdf)
... n can be written as a sum of two squares if and only if n = p21p22 . . . p2k q21+1q22+1 . . . q2l+1 where all the pi, qj are prime, and no qj is congruent to 3 mod 4. Proof. This follows immediately from the unique factorization property of Z[i] and the fact that norm is multiplicative. 3. Lag ...
... n can be written as a sum of two squares if and only if n = p21p22 . . . p2k q21+1q22+1 . . . q2l+1 where all the pi, qj are prime, and no qj is congruent to 3 mod 4. Proof. This follows immediately from the unique factorization property of Z[i] and the fact that norm is multiplicative. 3. Lag ...
... ______14) Which statement represents the Additive Identity? a) 12 ∙ 0 = 12 b) 12 + 0 = 12 c) 12 ∙ 1 = 12 d) 12 + 1 = 12 _____ 15) Which number is an integer but is not a whole number? a) ¼ b) 3 c) -5 d) 0 _____ 16) a) 13 b) – 41 c) – 13 d) 41 _____ 17) Which number is an integer but not a natural nu ...
Solutions to Some Review Problems for Exam 3 Recall that R∗, the
... Solution. This is not a subring, but the argument will use some things we haven’t seen. The easiest way to see it is to see that R is not closed under multiplication. ...
... Solution. This is not a subring, but the argument will use some things we haven’t seen. The easiest way to see it is to see that R is not closed under multiplication. ...
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.