# An introduction to the theory of field extensions by Samuel Moy

By Samuel Moy

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20) yields the eigenvalues (r 1)λ with multiplicity six when r D 2 and multiplicity eight when r D 3, in agreement with the detailed calculations presented in this section. 4 An n-Dimensional System with Diagonal Linear Part where J is a diagonal matrix with the entries λ 1 , λ 2 . . , λ n , and xD(x1 , x2 , . . , x n )T . Here, f r (x) is a real vector-valued function whose components are homogeneous polynomials of degree r. In other words, f r (x) belongs to the space H r , which is spanned by the vector-valued monomials x m e i D x1m 1 x2m 2 x nm 2 e i where m D (m 1 , m 2 , .

A has real eigenvalues and one of them is zero; that is, jAj D 0 and Tr(A) ¤ 0 where Tr(A) is the trace of A. 1 Introduction 2. A has purely imaginary eigenvalues; that is, jAj ¤ 0 and Tr(A) D 0 3. Both eigenvalues of A are zero but A is not the null matrix; that is, A¤0, jAj D 0 , and Tr(A) D 0 4. A D 0. The equality constraints are called degeneracy conditions. The number of degeneracy conditions indicates the level of degeneracy or codimension of the ﬁxed point (Dumortier, 1977; Guckenheimer and Holmes, 1983).

The quadratic terms have been ordered as O( ), whereas the cubic terms and the linear damping term have been ordered as O( 2 ). 107) and hence g 1 D 0 is justiﬁed a posteriori. 110), we conclude that the terms proportional to η and η 2 ηN are the only resonance terms and there are no near-resonance terms. 117) which is formally equivalent to that obtained by using the method of multiple scales. 131) in agreement with those obtained with the generalized method of averaging (Nayfeh, 1973). 85). 90).