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  2. Complex conjugate root theorem - Wikipedia

    en.wikipedia.org/wiki/Complex_conjugate_root_theorem

    The polynomial x 2 + 1 = 0 has roots ± i. Any real square matrix of odd degree has at least one real eigenvalue. For example, if the matrix is orthogonal, then 1 or −1 is an eigenvalue. The polynomial + has roots , +,, and thus can be factored as

  3. General number field sieve - Wikipedia

    en.wikipedia.org/wiki/General_number_field_sieve

    Consider the number field rings Z[r 1] and Z[r 2], where r 1 and r 2 are roots of the polynomials f and g. Since f is of degree d with integer coefficients, if a and b are integers, then so will be b d ·f(a/b), which we call r. Similarly, s = b e ·g(a/b) is an integer.

  4. RSA numbers - Wikipedia

    en.wikipedia.org/wiki/RSA_numbers

    The CPU time spent on finding these factors amounted to approximately 900 core-years on a 2.1 GHz Intel Xeon Gold 6130 CPU. Compared to the factorization of RSA-768, the authors estimate that better algorithms sped their calculations by a factor of 3–4 and faster computers sped their calculation by a factor of 1.25–1.67.

  5. Factorization - Wikipedia

    en.wikipedia.org/wiki/Factorization

    For example, 3 × 5 is an integer factorization of 15, and (x2)(x + 2) is a polynomial factorization of x 2 – 4. Factorization is not usually considered meaningful within number systems possessing division , such as the real or complex numbers , since any x {\displaystyle x} can be trivially written as ( x y ) × ( 1 / y ) {\displaystyle ...

  6. Factorization system - Wikipedia

    en.wikipedia.org/wiki/Factorization_system

    A weak factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that: [1] The class E is exactly the class of morphisms having the left lifting property with respect to each morphism in M. The class M is exactly the class of morphisms having the right lifting property with respect to each morphism in E.

  7. Special number field sieve - Wikipedia

    en.wikipedia.org/wiki/Special_number_field_sieve

    The SNFS works as follows. Let n be the integer we want to factor. As in the rational sieve, the SNFS can be broken into two steps: First, find a large number of multiplicative relations among a factor base of elements of Z/nZ, such that the number of multiplicative relations is larger than the number of elements in the factor base.

  8. Factor theorem - Wikipedia

    en.wikipedia.org/wiki/Factor_theorem

    In algebra, the factor theorem connects polynomial factors with polynomial roots. Specifically, if f ( x ) {\displaystyle f(x)} is a polynomial, then x − a {\displaystyle x-a} is a factor of f ( x ) {\displaystyle f(x)} if and only if f ( a ) = 0 {\displaystyle f(a)=0} (that is, a {\displaystyle a} is a root of the polynomial).

  9. Factorization of polynomials over finite fields - Wikipedia

    en.wikipedia.org/wiki/Factorization_of...

    The polynomial P = x 4 + 1 is irreducible over Q but not over any finite field. On any field extension of F 2, P = (x + 1) 4. On every other finite field, at least one of −1, 2 and −2 is a square, because the product of two non-squares is a square and so we have; If =, then = (+) ().