Variational methods for eigenvalue problems: An introduction by Sydney H. Gould PDF

By Sydney H. Gould

ISBN-10: 0802014046

ISBN-13: 9780802014047

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Let F be a finite, normal, separable extension of the field K. Suppose that the Galois group Gal(F/K) is isomorphic to D7 . Find the number of distinct subfields between F and K. How many of these are normal extensions of K? √ 5. Show that F = Q(i, 2) is normal over Q; find its Galois group over Q, and find all intermediate fields between Q and F . √ √ 6. Let F = Q( 2, 3 2). Find [F : Q] and prove that F is not normal over Q. 7. Find the order of the Galois group of x5 − 2 over Q. 4 Solvability by radicals Summary: We must first determine the structure of the Galois group of a polynomial of the form xn − a.

Thus we have the factorization x6 − 1 = x(x − 1)(x + 1)(x − 2)(x + 2)(x − 3)(x + 3). In solving the second half of the problem, looking for roots of x5 − 1 in Z11 is the same as looking for elements of order 5 in the multiplicative group Z× 11 . 10 states that the multiplicative group F × is cyclic if F is a finite field, so Z× 11 is cyclic of order 10. Thus it contains 4 elements of order 5, which means the x5 − 1 must split over Z11 . To look for a generator, we might as well start with 2. The powers of 2 are 22 = 4, 23 = 8, 24 = 5, 25 = −1, so 2 must be a generator.

5. The complex roots of the polynomial xn − 1 are the nth roots of unity. If we let α be the complex number α = cos θ + i sin θ, where θ = 2π/n, then 1, α, α2 , . , αn−1 are each roots of xn − 1, and since they are distinct they must constitute the set of all nth roots of unity. Thus we have n−1 (x − αk ) . n x −1= k=0 The set of nth roots of unity is a cyclic subgroup of C× of order n. Thus there are ϕ(n) generators of the group, which are the primitive nth roots of unity. If d|n, then any element of order d generates a subgroup of order d, which has ϕ(d) generators.

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Variational methods for eigenvalue problems: An introduction to the Weinstein method by Sydney H. Gould

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