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Edwin
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Does anyone know if Aut(F), the set of automorphisms of a field F, form a field under point-wise addition and multiplication of functions? If not, does it form a ring?
Inquisitively,
Edwin
Inquisitively,
Edwin
Why stop at binary operations, or associative ones? You might want to look into universal algebra. Or maybe model theory.Edwin said:Why did mathematicians not develop more abstract theory for two, three, or n associative binary operations on a set?
An automorphism of a field F is a bijective function from F to itself that preserves the field operations of addition and multiplication. In other words, an automorphism is a function that maps elements of the field to other elements in a way that the field properties remain unchanged.
Automorphisms can be seen as a type of symmetry in mathematics because they preserve the structure of a field. Just like how a symmetrical object stays unchanged after a transformation, a field remains unchanged after being mapped by an automorphism.
No, not all fields have automorphisms. For example, the field of real numbers has no non-trivial automorphisms, meaning that the identity function is the only automorphism of this field. However, some fields, such as finite fields, have a large number of automorphisms.
The automorphisms of a field F form a group called the Galois group of F. This group captures the symmetries of the field and is important in studying the solutions to polynomial equations over F.
Yes, automorphisms of a field F are unique. This means that for a given field, there is only one way to map its elements while preserving the field operations. However, different fields may have the same automorphisms if they are isomorphic, meaning they have the same structure.