Polynomial Irreducibility over Integers and Rationals

The question asks about the properties of a monic polynomial \(f(x)\) of degree \(n\) with integer coefficients, denoted as \(f(x) \in \mathbb{Z}[x]\). We need to evaluate the truthfulness of several statements regarding its irreducibility and roots.

Irreducibility in \(\mathbb{Z}[x]\) and \(\mathbb{Q}[x]\)

A key concept for relating irreducibility over integers and rationals is Gauss's Lemma. For a polynomial with integer coefficients, Gauss's Lemma states that it is reducible over the rational numbers \(\mathbb{Q}\) if and only if it is reducible over the integers \(\mathbb{Z}\), provided the polynomial is primitive. A primitive polynomial is one whose coefficients have a greatest common divisor (GCD) of 1.

Since \(f(x)\) is a monic polynomial, its leading coefficient is 1. The GCD of all coefficients of a monic polynomial must therefore be 1. This means a monic polynomial in \(\mathbb{Z}[x]\) is always primitive.

Applying Gauss's Lemma to our monic polynomial \(f(x) \in \mathbb{Z}[x]\):

• \(f(x)\) is reducible in \(\mathbb{Z}[x]\) if and only if it is reducible in \(\mathbb{Q}[x]\).

Taking the contrapositive of this statement:

• \(f(x)\) is irreducible in \(\mathbb{Z}[x]\) if and only if it is irreducible in \(\mathbb{Q}[x]\).

This equivalence directly addresses the first two options.

• Option 1: If \(f(x)\) is irreducible in \(\mathbb{Z}[x]\), then it is irreducible in \(\mathbb{Q}[x]\). This is part of the equivalence derived from Gauss's Lemma for monic (primitive) polynomials. Therefore, this statement is true.
• Option 2: If \(f(x)\) is irreducible in \(\mathbb{Q}[x]\), then it is irreducible in \(\mathbb{Z}[x]\). This is the other part of the equivalence derived from Gauss's Lemma for monic (primitive) polynomials. Therefore, this statement is also true.

Reducibility and Real Roots

Now let's consider the options related to real roots.

• Option 3: If \(f(x)\) is reducible in \(\mathbb{Z}[x]\), then it has a real root.

  If \(f(x)\) is reducible in \(\mathbb{Z}[x]\), it can be written as \(f(x) = g(x)h(x)\) where \(g(x), h(x) \in \mathbb{Z}[x]\) and \(\deg(g) \ge 1\), \(\deg(h) \ge 1\). However, the roots of \(g(x)\) and \(h(x)\) (and thus of \(f(x)\)) are not necessarily real. They could be complex numbers.

  Consider the monic polynomial \(f(x) = (x^2+1)(x^2+1) = x^4 + 2x^2 + 1\). This polynomial is in \(\mathbb{Z}[x]\) and is monic. It is reducible in \(\mathbb{Z}[x]\) because it factors as \((x^2+1)(x^2+1)\). The roots of \(f(x)\) are the roots of \(x^2+1=0\), which are \(x = \pm i\). These are complex roots, not real roots. Therefore, this statement is false.

• Option 4: If \(f(x)\) has a real root, then it is reducible in \(\mathbb{Z}[x]\).

  If \(f(x)\) has a real root \(r\), then \((x-r)\) is a factor of \(f(x)\) over the real numbers. For \(f(x)\) to be reducible in \(\mathbb{Z}[x]\), it must have a factor with integer coefficients (or rational coefficients, which for a monic polynomial implies integer coefficients for primitive factors). A real root \(r\) does not guarantee that \((x-r)\) or any factor corresponding to \(r\) is in \(\mathbb{Z}[x]\).

  Consider the monic polynomial \(f(x) = x^2 - 2\). This polynomial is in \(\mathbb{Z}[x]\) and is monic. Its roots are \(x = \pm \sqrt{2}\). These are real roots. However, \(x^2 - 2\) is irreducible over \(\mathbb{Z}[x]\) and \(\mathbb{Q}[x]\) because its roots \(\pm \sqrt{2}\) are irrational (by Eisenstein's Criterion with prime \(p=2\), or simply checking for rational roots \(\pm 1, \pm 2\)). Since it has real roots but is irreducible in \(\mathbb{Z}[x]\), this statement is false.

Conclusion

Based on the analysis using Gauss's Lemma and counterexamples:

• Option 1 is true.
• Option 2 is true.
• Option 3 is false.
• Option 4 is false.

Thus, the true statements are Option 1 and Option 2.