prove. Prove that the polynomial 1+x+x^ 2 +...+x^p-1 . where p is a prime number, is irreducible over the field of rational numbers.

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7. Let a be a root of X² - bX+c, for b, c integers such that b² - 4c < 0. Suppose that unique factorisation holds in Z[a]. Let m, n be relatively prime positive integers such that mn is expressible as x² + bxy + cy² for integers x and y. Show that both m and n are expressible in this form.