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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 3.8 Quotient Rings and Finite Fields ...
I need help with an aspect of the proof of Proposition 3.116
Proposition 3.116 and its proof reads as follows:View attachment 4696
View attachment 4697I need help with an aspect of the proof of (ii) above. In the proof of (ii) we read the following:
" ... ... $$k[x]/ (p(x))$$ is a field and, hence $$\text{ I am } \phi $$ is a field; that is, $$\text{ I am } \phi $$ is a subfield of $$K$$ containing $$k$$ and $$z$$, and so $$ k(z) \subseteq \text{ I am } \phi $$. ... ..."
Can someone please explain exactly why/how the above follows ... ...
... that is how, given that $$k[x]/ (p(x))$$ is a field, it then follows that $$\text{ I am } \phi $$ is a field; that is, $$\text{ I am } \phi $$ is a subfield of $$K$$ containing $$k$$ and $$z$$, and so $$k(z) \subseteq \text{ I am } \phi $$
Hope someone can help ... ...
PeterNOTE : The proof of (ii) above mentions Theorem 3.112, so in order for MHB readers to follow the above post, I am providing the statement of the theorem, as follows:View attachment 4698
I am currently focused on Section 3.8 Quotient Rings and Finite Fields ...
I need help with an aspect of the proof of Proposition 3.116
Proposition 3.116 and its proof reads as follows:View attachment 4696
View attachment 4697I need help with an aspect of the proof of (ii) above. In the proof of (ii) we read the following:
" ... ... $$k[x]/ (p(x))$$ is a field and, hence $$\text{ I am } \phi $$ is a field; that is, $$\text{ I am } \phi $$ is a subfield of $$K$$ containing $$k$$ and $$z$$, and so $$ k(z) \subseteq \text{ I am } \phi $$. ... ..."
Can someone please explain exactly why/how the above follows ... ...
... that is how, given that $$k[x]/ (p(x))$$ is a field, it then follows that $$\text{ I am } \phi $$ is a field; that is, $$\text{ I am } \phi $$ is a subfield of $$K$$ containing $$k$$ and $$z$$, and so $$k(z) \subseteq \text{ I am } \phi $$
Hope someone can help ... ...
PeterNOTE : The proof of (ii) above mentions Theorem 3.112, so in order for MHB readers to follow the above post, I am providing the statement of the theorem, as follows:View attachment 4698
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