Automorphisms and some maps that are bijective

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SUMMARY

This discussion focuses on demonstrating the bijectiveness of two functions, f_1 and f_2, in the context of abstract algebra. The function f_1 is identified as the identity map, which is inherently bijective. The second function, f_2, involves algebraic conjugates of roots from a polynomial and requires establishing a two-sided inverse to prove its bijectiveness. Participants emphasize the importance of understanding the representation of elements in the vector space V in terms of the roots of the polynomial.

PREREQUISITES
  • Understanding of abstract algebra concepts, particularly bijective functions.
  • Familiarity with polynomial roots and algebraic conjugates.
  • Knowledge of identity maps and their properties.
  • Basic understanding of vector spaces and field extensions.
NEXT STEPS
  • Study the properties of identity maps in abstract algebra.
  • Learn about algebraic conjugates and their role in polynomial equations.
  • Research the concept of two-sided inverses and their significance in proving bijectiveness.
  • Explore vector space representations and their implications in field extensions.
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Students and educators in abstract algebra, mathematicians focusing on polynomial functions, and anyone interested in understanding bijective mappings in algebraic structures.

Gott_ist_tot
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I am beginning abstract algebra and am having difficulty showing that some maps are bijective. It is a function f:V -> V where:

f_1 : ( 1 -> 1 and a -> a) and...
f_2 : ( 1 -> 1 and a -> a') where a and a' are the zeros of a polynomial.

f_1 seems trivial. Whatever you plug in you get again so I can not see how it could not be bijective. Going back to my topology class it is the identity map I believe. I am having difficulties getting f_2 where a' is the algebraic conjugate. Does anyone have suggestions about how to think about it. I just can not seem to get past the first hurdle of where to start.

Sorry for the formatting. I do not know how to make piecewise functions in tex. Thanks for any help in advance.
 
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I'm guessing V is the simple field extension of some field F generated by some a with minimal polynomial f(x), and a' is another root of f(x) which lies in V. Is this right? Please include this information next time.

If you've shown f_1 is the identity, you've shown its a bijection. Are you having trouble seeing why the identity is a bijection?

Have you been able to show what the elements in V look like in terms of a? I'm guessing you have if you've done the first part. If so, show there's a similar representation in terms of a', and use this to show f_2 has a two sided inverse, and therefore is a bijection.
 
Yes, I did understand the identity map. And your explanation of the two sided inverse makes sense. Thanks. I understand it.
 

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