Finite-Element Method Description

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• Fields : Let F be a finite field. Show that every element of F is the sum of two squares.

  58990 Finite Element Method Problem Let F be a finite field. Show that every element of F is the sum of two squares. (hint: given , show that and each have more than elements.

• Galois Theory : Use the method found in the proof below to find a primitive element for the extension Q(i, 5^(1/4)) over Q.

  58252 Method found in the proof below to find a primitive elements Use the method found in the proof below to find a primitive element for the extension Q(i, 5^(1/4)) over Q.

• Some Problems Involving Homomorphisms of Abelian Groups

  EDIT: G(p) = {x in G : |x| = p^k for some k greater than or equal to 0} ** Please see the attachment for the complete problem description ** 1. Let g be an element of G(p). Then |g| = p^k for some nonnegative integer k.

• Euclidean Algorithm

  Use the following definition of the order of an element in a finite group : The order of an element g in a finite group G is the least positive integer n such that gn=1. 5. 6. 7. 8. 9.

• Subgroup

  Every infinite group contains an element of infinite order. 5. Let G be a finite group in which every element has a square root; that is, for each x in G, there exists y in G with y=x2.

• Multipliciative groups

  Thus there are p 3 − p choices for the second basis element v2. Finally, any vector not in the span of {v1, v2} may be chosen as a third basis element.

• Galois Extension

  476474 Galois Extension ** Please see the attachment for a full problem description ** Suppose that K/F is a Galois extension, the Norm of alpha is an element of K is N_K/F := (please see the attached file) and trace is Tr_K/F(alpha) = (please see

• Infinite Groups

  That is, the coset is of finite order s (or a divisor of s). So every element of the quotient group Q/Z DOES have finite order. An infinite group is exemplified, such that every element has finite order

• Real-Life Experience with Maximizing or Minimizing and Application to Linear Programming; Geometric Discussion of the Simplex Method; Computer Programs for Linear Programming

  Therefore, Simplex method searches from one vertex to another nearby vertex until the optimal vertex is found. Since we have finite number of vertex, we promise the finite termination of the algorithm under certain pivot rule.