Prime Factorization

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Prime Factorization Prime Number A number with only two factors: ____ and itself Circle the prime numbers listed below 25 30 2 5 1 9 14 61 Composite Number A number that has more than 2 factors List five examples of composite numbers What kind of number is 0? What kind of number is 1? Every human has a unique fingerprint. Similarly, every COMPOSITE number has a unique "factorprint" called __________________________ Prime Factorization the factorization of a composite number into ____________ factors You can use a _________________ to find the prime factorization of any composite number. Ask yourself, "what Factorization two whole numbers 24 could I multiply together to equal the given number?" If the number is prime, do not put 1 x the number. Once you have all prime numbers, you are finished. Write your answer in exponential form. 24 Expanded Form (written as a multiplication of prime numbers) _______________________ Exponential Form (written with exponents) ________________________ Prime Factorization Ask yourself, "what two 36 numbers could I multiply together to equal the given number?" If the number is prime, do not put 1 x the number. Once you have all prime numbers, you are finished. Write your answer in both expanded and exponential forms. Prime Factorization Ask yourself, "what two 68 numbers could I multiply together to equal the given number?" If the number is prime, do not put 1 x the number. Once you have all prime numbers, you are finished. Write your answer in both expanded and exponential forms. Prime Factorization Ask yourself, "what two 120 numbers could I multiply together to equal the given number?" If the number is prime, do not put 1 x the number. Once you have all prime numbers, you are finished. Write your answer in both expanded and exponential forms. Prime Factorization 630.

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Algorithmic Factorization of Polynomials Over Number Fields
Rose-Hulman Institute of Technology Rose-Hulman Scholar Mathematical Sciences Technical Reports (MSTR) Mathematics 5-18-2017 Algorithmic Factorization of Polynomials over Number Fields Christian Schulz Rose-Hulman Institute of Technology Follow this and additional works at: https://scholar.rose-hulman.edu/math_mstr Part of the Number Theory Commons, and the Theory and Algorithms Commons Recommended Citation Schulz, Christian, "Algorithmic Factorization of Polynomials over Number Fields" (2017). Mathematical Sciences Technical Reports (MSTR). 163. https://scholar.rose-hulman.edu/math_mstr/163 This Dissertation is brought to you for free and open access by the Mathematics at Rose-Hulman Scholar. It has been accepted for inclusion in Mathematical Sciences Technical Reports (MSTR) by an authorized administrator of Rose-Hulman Scholar. For more information, please contact [email protected]. Algorithmic Factorization of Polynomials over Number Fields Christian Schulz May 18, 2017 Abstract The problem of exact polynomial factorization, in other words expressing a poly- nomial as a product of irreducible polynomials over some field, has applications in algebraic number theory. Although some algorithms for factorization over algebraic number fields are known, few are taught such general algorithms, as their use is mainly as part of the code of various computer algebra systems. This thesis provides a summary of one such algorithm, which the author has also fully implemented at https://github.com/Whirligig231/number-field-factorization, along with an analysis of the runtime of this algorithm. Let k be the product of the degrees of the adjoined elements used to form the algebraic number field in question, let s be the sum of the squares of these degrees, and let d be the degree of the polynomial to be factored; then the runtime of this algorithm is found to be O(d4sk2 + 2dd3).
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Algebraic Number Theory Summary of Notes
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