Exercise 6. Show that the following polynomials are irreducible over $Q$
1) $A = x^{4} - 4 x^{3} + 6$
2) $B = 4 x^{3} - 15 x^{2} + 60 x + 180$
3) $C = 8 x^{3} + 6 x^{2} - 9 x + 24$

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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 $p = 3$ : $3 ∤ 8, 3 | - 6, 3 | - 27, 3 | 9, 3^{2} ∤ 9$ , thus $C$ is irreducible over $Q$ .

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Step 1 ： $A$ : Apply Eisenstein's Criterion on $A = x^{4} - 4 x^{3} + 6$ with prime $p = 2$ : $2 ∤ 1, 2 | 4, 2 | 6, 2^{2} ∤ 6$ , thus $A$ is irreducible over $Q$ .

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 $p = 3$ : $3 ∤ 4, 3 | 15, 3 | 45, 3 | 60, 3^{2} ∤ 60$ , thus $B$ is irreducible over $Q$ .

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 $p = 3$ : $3 ∤ 8, 3 | - 6, 3 | - 27, 3 | 9, 3^{2} ∤ 9$ , thus $C$ is irreducible over $Q$ .

Exercise 6. Show that the following polynomials are irreducible over Q1) A=x4−4x3+62) B=4x3−15x2+60x+1803) C=8x3+6x2−9x+24

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Exercise 6. Show that the following polynomials are irreducible over $Q$
1) $A = x^{4} - 4 x^{3} + 6$
2) $B = 4 x^{3} - 15 x^{2} + 60 x + 180$
3) $C = 8 x^{3} + 6 x^{2} - 9 x + 24$