Exercise 6. Show that the following polynomials are irreducible over \( \mathbb{Q} \)
1) \( A=x^{4}-4 x^{3}+6 \)
2) \( B=4 x^{3}-15 x^{2}+60 x+180 \)
3) \( C=8 x^{3}+6 x^{2}-9 x+24 \)
Answer
\(C\): Substitute \(z = x - \frac{3}{4}\) to get \(C=8 z^{3} - 6 z^{2} - 27 z + 9\). Apply Eisenstein's Criterion with prime \(p=3\): \(3 \nmid 8, 3 | -6, 3 | -27, 3 | 9, 3^2 \nmid 9\), thus \(C\) is irreducible over \(\mathbb{Q}\).
Step 1 ：\(A\): Apply Eisenstein's Criterion on \(A=x^{4}-4 x^{3}+6\) with prime \(p=2\): \(2 \nmid 1, 2 | 4, 2 | 6, 2^2 \nmid 6\), thus \(A\) is irreducible over \(\mathbb{Q}\).
Step 2 ：\(B\): Substitute \(y = x + \frac{5}{4}\) to get \(B=4 y^{3} + 15 y^{2} + 45 y + 60\). Apply Eisenstein's Criterion with prime \(p=3\): \(3 \nmid 4, 3 | 15, 3 | 45, 3 | 60, 3^2 \nmid 60\), thus \(B\) is irreducible over \(\mathbb{Q}\).
Step 3 ：\(C\): Substitute \(z = x - \frac{3}{4}\) to get \(C=8 z^{3} - 6 z^{2} - 27 z + 9\). Apply Eisenstein's Criterion with prime \(p=3\): \(3 \nmid 8, 3 | -6, 3 | -27, 3 | 9, 3^2 \nmid 9\), thus \(C\) is irreducible over \(\mathbb{Q}\).
