without using any kind of label, comment one of Sylow's theorems and define Galois group of an extension.
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without using any kind of label, comment one of Sylow's theorems and define Galois group of an extension.
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- In maths ,What is the relationship between the degree of a field extension and the size of its Galois group?Let F be a finite field and let FC K be a finite extension. Then FC K is Galois with a cyclic Galois group.Find Aut(Z15) . Use the Fundamental Theorem of Abelian Groups to express this group as an external direct product of cyclic groups of prime power order. Please be clear with theorems, rules. Be legible.
- Determine the Galois group of x3 - 1 over Q and x3 - 2 over Q.Does the set of all positive rationals form a group with the multiplication operation? Either verify all of the conditions of the definition, or else show that one condition fails.Show that the inner direct product group is isomorphic to external direct sum group by constructing a function f:H1 x H2 x ... x Hn ---> H1 + H2 + ... + Hn via h1h2...hn ---> (h1, h2,...,hn), showing it is isomorphism. Please explain each step clearly. Thanks.
- Prove or disprove that {Z109, +, x} is a Galois Field?Give an example of the dihedral group of smallest order that contains a subgroup isomorphic to Z12 and a subgroup isomorphic to Z20. No need to prove anything, but explain your reasoning.Show that the Galois group of a polynomial of degree n has orderdividing n!