Find all units in the ring of Gaussian integers.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
100%

Could you explain how to show 10.14 in great detail? I also included a list of theorems and definitions in my textbook as a reference. Really appreciate your help!

10.14. Find all units in the ring of Gaussian integers.
Transcribed Image Text:10.14. Find all units in the ring of Gaussian integers.
Definition 10.3. Let R be an integral domain. Then a Euclidean function is a func-
tion ɛ from the set of nonzero elements of R to the nonnegative integers such that,
for all nonzero a, b e R,
we
have
1. ε(α) < ε(ab); and
2. there exist q,r e R such that a = bq +r, and either r = 0 or ɛ(r) < ɛ(b).
Definition 10.4. A Euclidean domain is an integral domain having a Euclidean
function.
Definition 10.5. Let R be a commutative ring with identity. If a, b e R, then we
say that a divides b, and write a|b, if there exists a c e R such that b = ac.
Definition 10.6. Let R be a Euclidean domain, and let a, b e R, not both zero. Then
a nonzero element d of R is said to be a greatest common divisor (or gcd) if
1. dļa and d|b; and
2. whenever c is an element of R satisfying c\a and c|b, we have ɛ(c) < ɛ(d).
Definition 10.7. Let F be a field and let f (x) and g(x) be polynomials in F[x], not
both the zero polynomial. By the gcd of f (x) and g(x) we mean a monic gcd. When
we write (f(x), g(x)), we mean specifically this monic gcd.
Theorem 10.4 (Euclidean Algorithm for Euclidean Domains). LetR be a
Euclidean domain. Take a, b e R with b 0. If b|a, then b is a gcd of a and
b. Otherwise, apply the division algorithm repeatedly. To wit, write
= bq1 + ri
a =
b = r¡92+ r2
ri = r293 + r3
Ik-2 = rk-1¶k + rk
rk-1 = rk¶k+1+0,
where all q; , r; e R and r; # 0, with &(r¡) < ɛ(b) and ɛ(r;) < ɛ(r;–1) for all j > 2.
Then rk is a gcd of a and b.
Corollary 10.3. Let R be a Euclidean domain. Take a, b e R with b + 0. Let d be
the gcd of a and b found in the preceding theorem. Then there exist u, v e R such
that d = au + bv.
Corollary 10.4. Let R be a Euclidean domain, and let a, b e R with b + 0. Let d
be the gcd of a and b found in Theorem 10.4. Then if c e R is a divisor of both a
and b, then c\d.
Definition 10.8. Let R be a commutative ring with identity. If a, b e R, then we
a and b are associates if there exists a unit u of R such that b
say
that
— ай.
Lemma 10.1. Let R be an integral domain. Then a and b are associates in R if and
only if a|b and b|a.
Theorem 10.5. Let R be a Euclidean domain. Take a, b e R, not both 0. Let d
be any gcd of a and b. Thenc e R is a gcd of a and b if and only if c and d are
associates.
Theorem 10.6. Let R be a Euclidean domain. Take a, b e R, not both 0. Let d be
a gcd of a and b. Then there exist u, v e R such that d = au + bv.
Theorem 10.7. Let R be a Euclidean domain. Take a, b e R, not both 0. Then the
following are equivalent for an element d of R:
1. d is a gcd of a and b; and
2. d|a, d\b, and if c\a and c\b, then c\d.
Transcribed Image Text:Definition 10.3. Let R be an integral domain. Then a Euclidean function is a func- tion ɛ from the set of nonzero elements of R to the nonnegative integers such that, for all nonzero a, b e R, we have 1. ε(α) < ε(ab); and 2. there exist q,r e R such that a = bq +r, and either r = 0 or ɛ(r) < ɛ(b). Definition 10.4. A Euclidean domain is an integral domain having a Euclidean function. Definition 10.5. Let R be a commutative ring with identity. If a, b e R, then we say that a divides b, and write a|b, if there exists a c e R such that b = ac. Definition 10.6. Let R be a Euclidean domain, and let a, b e R, not both zero. Then a nonzero element d of R is said to be a greatest common divisor (or gcd) if 1. dļa and d|b; and 2. whenever c is an element of R satisfying c\a and c|b, we have ɛ(c) < ɛ(d). Definition 10.7. Let F be a field and let f (x) and g(x) be polynomials in F[x], not both the zero polynomial. By the gcd of f (x) and g(x) we mean a monic gcd. When we write (f(x), g(x)), we mean specifically this monic gcd. Theorem 10.4 (Euclidean Algorithm for Euclidean Domains). LetR be a Euclidean domain. Take a, b e R with b 0. If b|a, then b is a gcd of a and b. Otherwise, apply the division algorithm repeatedly. To wit, write = bq1 + ri a = b = r¡92+ r2 ri = r293 + r3 Ik-2 = rk-1¶k + rk rk-1 = rk¶k+1+0, where all q; , r; e R and r; # 0, with &(r¡) < ɛ(b) and ɛ(r;) < ɛ(r;–1) for all j > 2. Then rk is a gcd of a and b. Corollary 10.3. Let R be a Euclidean domain. Take a, b e R with b + 0. Let d be the gcd of a and b found in the preceding theorem. Then there exist u, v e R such that d = au + bv. Corollary 10.4. Let R be a Euclidean domain, and let a, b e R with b + 0. Let d be the gcd of a and b found in Theorem 10.4. Then if c e R is a divisor of both a and b, then c\d. Definition 10.8. Let R be a commutative ring with identity. If a, b e R, then we a and b are associates if there exists a unit u of R such that b say that — ай. Lemma 10.1. Let R be an integral domain. Then a and b are associates in R if and only if a|b and b|a. Theorem 10.5. Let R be a Euclidean domain. Take a, b e R, not both 0. Let d be any gcd of a and b. Thenc e R is a gcd of a and b if and only if c and d are associates. Theorem 10.6. Let R be a Euclidean domain. Take a, b e R, not both 0. Let d be a gcd of a and b. Then there exist u, v e R such that d = au + bv. Theorem 10.7. Let R be a Euclidean domain. Take a, b e R, not both 0. Then the following are equivalent for an element d of R: 1. d is a gcd of a and b; and 2. d|a, d\b, and if c\a and c\b, then c\d.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

Blurred answer
Knowledge Booster
Basics of Inferential Statistics
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,