Irreducible polynomials

In summary, to show that x^2+1 is irreducible over the integers, it is enough to show that it can only be factored into (x-i)(x+i), has no roots in the integers, and thus cannot be reduced. This is because if it were reducible, it would have two linear factors with integer coefficients, but it has no real solutions. However, it should be noted that strictly speaking, the reason is that there are no integers satisfying x^2+1=0.
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If i have to show a polynomial x^2+1 is irreduceable over the integers, is it enough to show that X^2 + 1 can only be factored into (x-i)(x+i), therefore has no roots in the integers, and is subsequently irreduceable?
 
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I wouldn't drag imaginary numbers into this. If x^2+1 is reducible over the integers then it would split into two linear factors with integer coefficients. Can you argue why that can't happen?
 
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is it because x^2+1 has no real solutions?
 
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Strictly speaking, because the problem said "irreduceable over the integers", it is because there are no integers satisfying [itex]x^2+1= 0[/itex]. Of course, since the integers are a subset of the real numbers yours is a sufficient answer. But your teacher might call your attention to the difference by (on a test, perhaps) asking you to show that [itex]x^2- 2[/itex] is irreduceable over the integers.
 

1. What are irreducible polynomials?

Irreducible polynomials are polynomials that cannot be factored into smaller polynomials with coefficients from the same field. In other words, they are polynomials that do not have any factors other than 1 and itself.

2. How do you determine if a polynomial is irreducible?

There are several methods for determining if a polynomial is irreducible. One method is to check if the polynomial has any linear factors (polynomials of degree 1) by using the factor theorem. If it does not have any linear factors, then it may be irreducible. Another method is to use the Eisenstein's criterion, which states that if a polynomial satisfies certain conditions, then it is irreducible.

3. What is the importance of irreducible polynomials?

Irreducible polynomials are important in many areas of mathematics, particularly in algebra and number theory. They are used in the construction of finite fields and in the factorization of polynomials over a given field. They also have applications in coding theory, cryptography, and other areas of computer science.

4. Can a polynomial have multiple irreducible factors?

Yes, a polynomial can have multiple irreducible factors. For example, the polynomial x^2 + 1 has two irreducible factors, x + 1 and x - 1. However, if a polynomial has only one irreducible factor, then it is called a prime polynomial.

5. Are all polynomials irreducible?

No, not all polynomials are irreducible. For example, the polynomial x^2 + 2x + 1 can be factored into (x + 1)^2, so it is not irreducible. However, all polynomials with degree 1 (linear polynomials) are irreducible.

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