Factorisation of cyclotomic polynomials

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In summary, the speaker confirms that x^16 + 1 is indeed irreducible over the rationals. They then mention their approach to factoring polynomials and how the given hint may be used to alter the roots of an equation. They suggest using this hint creatively to solve the given problem.
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catcherintherye
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x^16 + 1 is irreducible over the rationals, correct?...


...also I am required to factorise the following polynomials

2) i) x^5 + 3x^4 + 2x^3 + x^2 -7

ii) x^5 + 10x^4 + 13x^3 -25x^2 -68x -60


now I would usually approach this using the factor theorem to find a factor and then divide by this factor and continue, however in the question I am told as a hint that I am to try substituting x-> x+h h=+-1, +-2, why all this is necessary i canot think?...
 
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  • #2
"x^16 + 1 is irreducible over the rationals, correct?..."

yup..

As for your hint, that is usually used to alter the roots of an equation. eg of the roots of a polynomial of degree 3 in x, is alpha, beta and gamma, subbing in x+1 will get you a polynomial that has roots alpha-1, beta -1 , gamma -1 etc etc.

Maybe that's a hint your ment to use creatively?
 

FAQ: Factorisation of cyclotomic polynomials

What are cyclotomic polynomials and why do we need to factorize them?

Cyclotomic polynomials are a special type of polynomial that arise in the study of complex numbers, specifically roots of unity. They are important because they have many connections to other areas of mathematics such as number theory and algebraic geometry. Factorizing them helps us understand their properties and relationships with other polynomials.

How do we factorize cyclotomic polynomials?

There are several methods for factorizing cyclotomic polynomials, including the use of cyclotomic fields and the use of modular arithmetic. One of the most common methods is to use the factorization formula for cyclotomic polynomials, which involves the use of complex roots of unity.

What is the significance of the factors in the factorization of cyclotomic polynomials?

The factors in the factorization of cyclotomic polynomials have important applications in number theory, algebraic geometry, and other areas of mathematics. They also provide insights into the structure of the polynomial and its relationship with other polynomials.

Can all cyclotomic polynomials be factorized?

Yes, all cyclotomic polynomials can be factorized into linear factors. This is a consequence of the fundamental theorem of algebra, which states that every polynomial of degree n has n complex roots (counting multiplicity). Since cyclotomic polynomials have degree n, they can be factorized into n linear factors.

Are there any applications of factorization of cyclotomic polynomials in other fields of science?

Yes, the factorization of cyclotomic polynomials has applications in various fields of science, including cryptography, coding theory, and signal processing. In these fields, the factorization of cyclotomic polynomials is used to construct efficient algorithms and to solve certain problems.

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