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I am reading Anderson and Feil - A First Course in Abstract Algebra.
I am currently focused on Ch. 42: Field Extensions and Kronecker's Theorem ...
I need some help with an aspect of the proof of Theorem 42.1 ( Kronecker's Theorem) ...
Theorem 42.1 and its proof read as follows:
https://www.physicsforums.com/attachments/6565
https://www.physicsforums.com/attachments/6566In the above text by Anderson and Feil we read the following:
" ... ... We show that there is an isomorphism from into by considering the function defined by , where . ... ... "The authors show that is one-to-one or injective but do not show that is onto or surjective ...
My question is ... how do we know that is surjective ...
... for example if a polynomial in , say , is degree 5, and is degree 3 then dividing by gives a polynomial remainder of degree 2 ... then will not be of the form where ... ... and so it seems that is not surjective ... since the coset of is not of the form where ...
... ?Obviously my thinking is somehow mistaken ...... can anyone help by demonstrating that is surjective ... and hence (given that Anderson and Feil have demonstrated it is injective) an isomorphism ...Help will be appreciated ...
Peter
I am currently focused on Ch. 42: Field Extensions and Kronecker's Theorem ...
I need some help with an aspect of the proof of Theorem 42.1 ( Kronecker's Theorem) ...
Theorem 42.1 and its proof read as follows:
https://www.physicsforums.com/attachments/6565
https://www.physicsforums.com/attachments/6566In the above text by Anderson and Feil we read the following:
" ... ... We show that there is an isomorphism from
My question is ... how do we know that
... for example if a polynomial in
... ?Obviously my thinking is somehow mistaken ...... can anyone help by demonstrating that
Peter
Last edited: