*** Let R be a Euclidean Domain with norm function given by N. Then R is a Let R be a Euclidean Domain with norm function given by N. Let I be an ideal of R. Define S := {N(i): i E I – {0}} .: S has a least element. Define d := argmin(S) We now claim that I = (d) and prove it: Let x E I. Note tha d # 0. x = dq +r for some d, r ER such that either r = 0R or N(r) < N(d). N(r) 4 N(d) :. r = OR %3D .. d|x : x € (d) :. IC (d) Note that (d) CI :. I = (d) %3D Therefore R is a PID.
*** Let R be a Euclidean Domain with norm function given by N. Then R is a Let R be a Euclidean Domain with norm function given by N. Let I be an ideal of R. Define S := {N(i): i E I – {0}} .: S has a least element. Define d := argmin(S) We now claim that I = (d) and prove it: Let x E I. Note tha d # 0. x = dq +r for some d, r ER such that either r = 0R or N(r) < N(d). N(r) 4 N(d) :. r = OR %3D .. d|x : x € (d) :. IC (d) Note that (d) CI :. I = (d) %3D Therefore R is a PID.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Justify each step that makes a claim no matter how trivial.
Transcribed Image Text:**Euclidean Domains and Principal Ideal Domains**
Let \( R \) be a Euclidean Domain with a norm function given by \( N \). Then \( R \) is a Principal Ideal Domain (PID).
### Proof:
1. **Assumptions:**
- Let \( R \) be a Euclidean Domain with a norm function given by \( N \).
- Let \( I \) be an ideal of \( R \).
2. **Definitions:**
- Define \( S := \{N(i) : i \in I - \{0\}\} \).
- Since \( S \) is a set of norms of non-zero elements of \( I \), \( S \) has a least element.
3. **Definitions and Claims:**
- Define \( d := \text{argmin}(S) \). This means \( d \) is such that \( N(d) \) is the smallest element in \( S \).
- We claim that \( I = \langle d \rangle \) and aim to prove it.
4. **Proof:**
- Consider any element \( x \in I \).
- Note that \( d \neq 0 \).
- Express \( x = dq + r \) for some \( d, r \in R \) such that either \( r = 0_R \) or \( N(r) < N(d) \).
- If \( r \neq 0 \), then it would contradict the minimality of \( N(d) \).
- Therefore, \( r = 0_R \).
- This implies \( d \mid x \) (i.e., \( d \) divides \( x \)).
- Consequently, \( x \in \langle d \rangle \).
- It follows that \( I \subseteq \langle d \rangle \).
5. **Conclusion:**
- Note that \( \langle d \rangle \subseteq I \).
- Therefore, \( I = \langle d \rangle \), concluding the proof.
- Hence, \( R \) is a Principal Ideal Domain (PID).
This proof demonstrates the property of Euclidean Domains that ensures they are Principal Ideal Domains, highlighting the significance of the norm function in determining the structure of ideals in such domains.
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