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Chapter 10: Q10.2.2E (page 341)

Suppose p is an irreducible element in an integral domain R such that whenever , then or . If , prove that p divides at least one.

Short Answer

Expert verified

It is proved that p divides at least one.

Step by step solution

01

Referring to the Theorem 10.15

Theorem 10.15

Let P be an irreducible element in a unique factorization domain R. If then or .

Given that p is an irreducible element in an integral domain R.

02

Proving that p divides at least one

As given in the question, whenever then or .

Therefore, by Theorem 10.15, it is a unique factorization domain R.

So, according to the theorem, if then or or

Therefore, for any , we can conclude that p divides any one of .

Hence, it is proved that p divides at least one .

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Most popular questions from this chapter

If u andv are units, prove that u and v are associates.

(a): Verify that each of $5 + \sqrt{2}$ , $2 - \sqrt{2}$ , $11 - 7 \sqrt{2}$ and $2 + \sqrt{2}$ is irreducible in $ℤ [\sqrt{2}]$ .

(b) Explain why the fact that

$(5 + \sqrt{2}) (2 - \sqrt{2}) = (11 - 7 \sqrt{2}) (2 + \sqrt{2})$

does not contradict unique factorization in .

LetR be any integral domain and $p \in R$ . Prove that p is irreducible in R if and only if the constant polynomial p is irreducible in $R [x]$ . [Hint: Corollary 4.5 may be helpful.]

(a). Show that the only divisors of $x$ in $ℚ_{ℤ} [x]$ are the integers (constant polynomial) and the first-degree polynomial of the form $\frac{1}{n} x$ with $0 \neq n \in ℤ$ .

(b) For each nonzero $n \in ℤ$ , show that the polynomial $\frac{1}{n} x$ is not irreducible in $ℚ_{ℤ} [x]$ .

(c) Show that $x$ cannot be written as a finite product of irreducible elements in $ℚ_{ℤ} [x]$ .

A ring $R$ is said to satisfy the descending chain condition (DCC) on ideals if whenever $I_{1} \supseteq I_{2} \supseteq I_{3} \dots$ is a chain of ideals in $R$ , then there is an integer $n$ such that $I_{j} = I_{n}$ for all $j \geq n$ .

(a) Show that $ℤ$ does not satisfy the DCC.

(b) Show that an integral domain $R$ is a field if and only if $R$ satisfies the DCC.

See all solutions

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