Determine, with proof, which of the following polynomials are irreducible over \(\mathbb Q\):
\(x^2+3\);
\(x^2-169\);
\(x^3+x^2+x+1\);
\(x^3+2x^2+3x+4\).
Solution
1. A polynomial in \(\QQ[x]\) of degree \(2\) or \(3\) is reducible over\(\QQ\) if and only if it has a root in \(\QQ\).
If such a polynomial has a root \(r\in\QQ\), then the factor theorem gives a nontrivial factor \(x-r\), so the polynomial is reducible. Conversely, suppose a polynomial of degree \(2\) or \(3\) factors nontrivially over \(\QQ\). The positive degrees of its two factors add to \(2\) or \(3\), so at least one factor has degree \(1\). A linear factor over \(\QQ\) has a root in \(\QQ\), which is also a root of the original polynomial.
2. Item 1,\[ x^2+3, \]is irreducible over \(\QQ\).
For every \(r\in\QQ\subset\RR\), \[ r^2+3>0. \] Thus the polynomial has no rational root. Step 1 implies that it is irreducible.
3. Item 2,\[ x^2-169, \]is reducible over \(\QQ\).
One has \[ x^2-169 = (x-13)(x+13), \] a nontrivial factorization in \(\QQ[x]\).
4. Item 3,\[ x^3+x^2+x+1, \]is reducible over \(\QQ\).
Grouping terms gives \[ \begin{aligned} x^3+x^2+x+1 &= x^2(x+1)+(x+1)\\ &= (x+1)(x^2+1). \end{aligned} \] Both factors have positive degree.
5. If\[ P(x)=x^3+2x^2+3x+4 \]has a rational root, then that root belongs to\[ \{-4,-2,-1,1,2,4\}. \]
Suppose \(a/b\in\QQ\) is a root, where \(a,b\in\ZZ\), \(b>0\), and \(\gcd(a,b)=1\). Multiplying \[ \left(\frac ab\right)^3 +2\left(\frac ab\right)^2 +3\left(\frac ab\right) +4 =0 \] by \(b^3\) gives \[ a^3+2a^2b+3ab^2+4b^3=0. \] Hence \(b\mid a^3\). Since \(\gcd(a,b)=1\), this forces \(b=1\), so the root is an integer \(a\). The equation \[ a(a^2+2a+3)=-4 \] then shows that \(a\) divides \(4\). Therefore \[ a\in\{-4,-2,-1,1,2,4\}. \]
6. Item 4,\[ x^3+2x^2+3x+4, \]is irreducible over \(\QQ\).
For the six possible rational roots from step 5, \[ \begin{aligned} P(-4)&=-40,& P(-2)&=-2,& P(-1)&=2,\\ P(1)&=10,& P(2)&=26,& P(4)&=112. \end{aligned} \] None is zero. Thus the polynomial has no rational root, and step 1 implies that it is irreducible.
7. Therefore the irreducible polynomials are exactly\[ \boxed{x^2+3} \qquad\text{and}\qquad \boxed{x^3+2x^2+3x+4}. \]
Step 2 proves item 1 irreducible, step 3 proves item 2 reducible, step 4 proves item 3 reducible, and step 6 proves item 4 irreducible.
8. Q.E.D.
Step 7 gives the requested classification.