NAP PROBLEM SET #1, SOLUTIONS 1. We follow the procedure in
... integers. If m is even, say, m = 2`, then g = (x` )2 , contrary to our assumption. Thus m is odd, and similarly n is odd. Therefore m + n is even, say, m + n = 2q. Now we have gh = xm xn = xm+n = x2q = (xq )2 . This shows that gh is a square. (Note that the proof did not use the hypothesis that |G| ...
... integers. If m is even, say, m = 2`, then g = (x` )2 , contrary to our assumption. Thus m is odd, and similarly n is odd. Therefore m + n is even, say, m + n = 2q. Now we have gh = xm xn = xm+n = x2q = (xq )2 . This shows that gh is a square. (Note that the proof did not use the hypothesis that |G| ...
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... of polynomials are non-negative integers. If f and g belong to K[X] and the latter of them is not the zero polynomial, then, as is well known, the long division f /g gives two unique polynomials q and r in K[X] such that f = qg + r, where ν(r) < ν(g) or r is the zero polynomial. The second property ...
... of polynomials are non-negative integers. If f and g belong to K[X] and the latter of them is not the zero polynomial, then, as is well known, the long division f /g gives two unique polynomials q and r in K[X] such that f = qg + r, where ν(r) < ν(g) or r is the zero polynomial. The second property ...
Quadratic equations and complex numbers
... The quantity under the radical, b2 - 4ac, is called the discriminant (D). Assuming a, b, c to be real numbers: If D > 0, the two roots of the equation are distinct real numbers. If D = 0, the equation has a real, double root (the roots are equal and real). If D < 0, the roots of the equation are two ...
... The quantity under the radical, b2 - 4ac, is called the discriminant (D). Assuming a, b, c to be real numbers: If D > 0, the two roots of the equation are distinct real numbers. If D = 0, the equation has a real, double root (the roots are equal and real). If D < 0, the roots of the equation are two ...
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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.