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I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ...
I am currently focused on Section 3.3 Polynomials ...
I need help with an aspect of the proof of Proposition 3.29 concerning elements of the function field $$k(x)$$.
Rotman's definition of a function field over $$k$$ and his Proposition 3.29 and its proof read as follows:View attachment 4695
In the proof of Proposition 3.29 displayed above, we read the following:
" ... ... elements in $$k(x)$$ have the form $$f(x) {g(x)}^{-1}$$ ... ... "I am perplexed by the above statement as it includes the polynomial $${g(x)}^{-1}$$ and I thought the only inverses in $$k[x]$$ were the constant polynomials ... and that no other inverses existed in $$k[x]$$ ... so how are we to make sense of the above statement ... unless we just regard $$f(x) {g(x)}^{-1}$$ as the result of $$f(x)$$ divided by $$g(x)$$ ...
Can someone please clarify the above issue ...
Peter
I am currently focused on Section 3.3 Polynomials ...
I need help with an aspect of the proof of Proposition 3.29 concerning elements of the function field $$k(x)$$.
Rotman's definition of a function field over $$k$$ and his Proposition 3.29 and its proof read as follows:View attachment 4695
In the proof of Proposition 3.29 displayed above, we read the following:
" ... ... elements in $$k(x)$$ have the form $$f(x) {g(x)}^{-1}$$ ... ... "I am perplexed by the above statement as it includes the polynomial $${g(x)}^{-1}$$ and I thought the only inverses in $$k[x]$$ were the constant polynomials ... and that no other inverses existed in $$k[x]$$ ... so how are we to make sense of the above statement ... unless we just regard $$f(x) {g(x)}^{-1}$$ as the result of $$f(x)$$ divided by $$g(x)$$ ...
Can someone please clarify the above issue ...
Peter
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