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An algebraic number field is a finite extension of Q. Let K be an algebraic number field. Each element 2 K satisfies an equation n + a1n1 + ::: + an with coefficients a1; :::; an 2 Q and is an algebraic integer if it satisfies such an equation with coefficients a1; :::; an 2 Z. algebraic integers form a subring.Let K be an algebraic number field of degree n. Let OK denote the ring of integers of K. The field K is said to possess a power basis if there exists an element 2 OK such that OK = Z + Z + + Zn1. A field having a power basis is called monogenic. In this thesis we consider that there exists infinitely many dihedral quintic fields with a power basis that is shown by lavallee [18]  

LetK be a biquadratic field. M.-N. Gras and F. Tanoe gave a necessary and sufficient condition that K is monogenic by using a diophantine equation of degree 4 [13]. We consider algebraic extension fields of higher degree. Let F be a Galois extension field over the rationals Q whose Galois group is 2-elementary Abelian. then we shall prove that F of degree graeter than 8, is monogenic if and only if F being field of n'th primitive root of unity under a suitable condition for the case of degree 8