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I am reading Chapter 2: Commutative Rings in Joseph Rotman's book, Advanced Modern Algebra (Second Edition).
I am currently focussed on Theorem 2.64 [pages 115 - 116] concerning irreducibility.
I need help to the proof of this theorem.
Theorem 2.64 and its proof read as follows:https://www.physicsforums.com/attachments/2695
https://www.physicsforums.com/attachments/2696In the above text: Rotman writes:
" ... ... Suppose that $$ f(x) $$ factors in $$ \mathbb{Z} [x] $$; say $$ f(x) = g(x)h(x) $$, where $$ deg(g) \lt deg(f) $$ and $$ deg(h) \lt deg(f) $$. Now $$ \overline{f}(x) = \overline{g}(x) \overline{h}(x) $$ so that $$ deg( \overline{f}) = deg( \overline{g}) + deg( \overline{h}). $$ Now, $$ \overline{f}(x) $$ is monic because f(x) is and so $$ deg( \overline{f}) = deg(f) $$. Thus both $$ deg( \overline{g}) \text{ and } deg( \overline{h}) $$ have degrees less than $$ deg( \overline{f}) $$ ... ..."
My question is as follows:
How does (on what grounds) does Rotman reach the conclusion that both $$ deg( \overline{g}) \text{ and } deg( \overline{h}) $$ have degrees less than $$ deg( \overline{f}) $$?
Indeed, what is preventing one of $$ deg( \overline{g}) , deg( \overline{h}) $$ from being equal to 1 and hence having degree zero and the other having degree n? (Having one of them having degree 0 and being equal to a unit other than 1 would, of course, contradict the monic nature of f, but one of them being equal to 1 seems an open possibility)
Hope someone can help clarify the above issue.
Peter
I am currently focussed on Theorem 2.64 [pages 115 - 116] concerning irreducibility.
I need help to the proof of this theorem.
Theorem 2.64 and its proof read as follows:https://www.physicsforums.com/attachments/2695
https://www.physicsforums.com/attachments/2696In the above text: Rotman writes:
" ... ... Suppose that $$ f(x) $$ factors in $$ \mathbb{Z} [x] $$; say $$ f(x) = g(x)h(x) $$, where $$ deg(g) \lt deg(f) $$ and $$ deg(h) \lt deg(f) $$. Now $$ \overline{f}(x) = \overline{g}(x) \overline{h}(x) $$ so that $$ deg( \overline{f}) = deg( \overline{g}) + deg( \overline{h}). $$ Now, $$ \overline{f}(x) $$ is monic because f(x) is and so $$ deg( \overline{f}) = deg(f) $$. Thus both $$ deg( \overline{g}) \text{ and } deg( \overline{h}) $$ have degrees less than $$ deg( \overline{f}) $$ ... ..."
My question is as follows:
How does (on what grounds) does Rotman reach the conclusion that both $$ deg( \overline{g}) \text{ and } deg( \overline{h}) $$ have degrees less than $$ deg( \overline{f}) $$?
Indeed, what is preventing one of $$ deg( \overline{g}) , deg( \overline{h}) $$ from being equal to 1 and hence having degree zero and the other having degree n? (Having one of them having degree 0 and being equal to a unit other than 1 would, of course, contradict the monic nature of f, but one of them being equal to 1 seems an open possibility)
Hope someone can help clarify the above issue.
Peter