MHB Proving Irreducibility of x^4 −7 Using Polynomial Theorems

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The polynomial x^4 - 7 is irreducible over the rational numbers Q because it cannot be factored into polynomials of lower degree with rational coefficients. The Eisenstein Criterion is often cited in such discussions, but it does not apply directly here. Instead, one can use the fact that x^4 - 7 has no rational roots, as confirmed by the Rational Root Theorem. Additionally, it can be shown that the polynomial does not factor into two quadratic polynomials with rational coefficients. Thus, x^4 - 7 is confirmed as irreducible over Q.
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Explain why the polynomial x^4 −7 is irreducible over Q, quoting any theorems you use.
 
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Which theorems related to irreducibility do you know?
 

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