Is a^2 - 6a + 12 Irreducible Over the Integers?

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In summary, "Irreducible Over the Integers" is a mathematical concept where a polynomial cannot be factored into smaller polynomials with integer coefficients. This concept differs from irreducibility over other sets of numbers, as a polynomial may be irreducible over the integers but reducible over the real or complex numbers. Studying irreducible polynomials over the integers has many applications in mathematics, including number theory, algebra, and cryptography. However, not all polynomials can be reduced over the integers, as some may remain irreducible, such as x^2 + 1. To determine if a polynomial is irreducible over the integers, there are methods such as Eisenstein's criterion, the rational root theorem, and the Berlek
  • #1
mathdad
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In the textbook, the author showed that 8 + (a - 2)^3 factors out to be a(a^2 - 6a + 12).

The author goes on to say "...the expression a^2 - 6a + 12 is irreducible over the integers."

What does the author means by the statement?
 
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  • #2
It simply means that the polynomial $a^2-6a+12$ cannot be factored into the product of two polynomials with coefficients that are integers. :D
 
  • #3
Can you provide me with a few more irreducible examples?
 

FAQ: Is a^2 - 6a + 12 Irreducible Over the Integers?

What is "Irreducible Over the Integers"?

"Irreducible Over the Integers" refers to a mathematical concept where a polynomial cannot be factored into smaller polynomials with integer coefficients.

How is "Irreducible Over the Integers" different from being irreducible over a different set of numbers?

The concept of irreducibility depends on the set of numbers being considered. A polynomial may be irreducible over the integers, but reducible over the real or complex numbers.

What is the importance of studying "Irreducible Over the Integers"?

Studying irreducible polynomials over the integers has many applications in mathematics, including number theory, algebra, and cryptography.

Can all polynomials be reduced over the integers?

No, not all polynomials can be reduced over the integers. For example, the polynomial x^2 + 1 is irreducible over the integers.

How can "Irreducible Over the Integers" be determined?

There are a few methods for determining if a polynomial is irreducible over the integers, such as the Eisenstein's criterion, the rational root theorem, and the Berlekamp's algorithm.

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