Question

Which of the following polynomials is irreducible over {Q} ?





Question 4Answer

a.
x
^
2

+

1


b.
x
^
2

-

4


c.
x
^
2

-

1


d.
x
^
2

+

2
x

+

1

Answer 1

To determine which of the given polynomials is irreducible over the field of rational numbers \(\mathbb{Q}\), we need to check if each polynomial can be factored into the product of two non-trivial polynomials with coefficients in \(\mathbb{Q}\):

1. **\(x^2 + 1\)**: This polynomial has no rational roots (as \(x^2 + 1 = 0\) leads to \(x^2 = -1\) which has no real solutions), and it cannot be factored into products of linear polynomials with rational coefficients. Hence, it is irreducible over \(\mathbb{Q}\).

2. **\(x^2 - 4\)**: This polynomial can be factored as \((x - 2)(x + 2)\), which has rational coefficients. Therefore, it is reducible over \(\mathbb{Q}\).

3. **\(x^2 - 1\)**: This polynomial can be factored as \((x - 1)(x + 1)\), which has rational coefficients as well. Thus, it is reducible over \(\mathbb{Q}\).

4. **\(x^2 + 2x + 1\)**: This polynomial can be factored as \((x + 1)(x + 1)\) or \((x + 1)^2\). It is also reducible over \(\mathbb{Q}\).

After analyzing all options, the only irreducible polynomial over \(\mathbb{Q}\) is:

**\(x^2 + 1\)** (option a).