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Hello!! In a recent post Julmath presented an example that solve part (a) of Problem 7.1. Then I will focus in solve part (b).
Solution of part (b)
Solution of part (b)
Take the polynomial . It is easy to see that it has some positive root . Moreover, it is irreducible and therefore it is the minimal polynomial of . Consider the monoid . We will prove that is atomic by arguing that .
We first assume that . Then there exists such that . Hence is a root of . Since is the minimal polynomial of there exists some polynomial such that . It is easy to see that the constant coefficient of is . However, Gauss lemma ensures that which is a contradiction.
Let us suppose now that for some . Since , we can infer that for every . Then, after dividing both sides of the previous equation by , we obtain that is not an atom, which is a contradiction. Therefore we conclude that is atomic
To see that does not satisfies the ACCP observe that is an ascending chain of principal ideals that does not stabilizes. In fact, since .
Since belongs to the family of polynomials that I presented in my last post, we already know that is a rank- monoid such that . This conclude this part of the prove.
We first assume that . Then there exists such that . Hence is a root of . Since is the minimal polynomial of there exists some polynomial such that . It is easy to see that the constant coefficient of is . However, Gauss lemma ensures that which is a contradiction.
Let us suppose now that for some . Since , we can infer that for every . Then, after dividing both sides of the previous equation by , we obtain that is not an atom, which is a contradiction. Therefore we conclude that is atomic
To see that does not satisfies the ACCP observe that is an ascending chain of principal ideals that does not stabilizes. In fact, since .
Since belongs to the family of polynomials that I presented in my last post, we already know that is a rank- monoid such that . This conclude this part of the prove.