Find all units in the ring of Gaussian integers.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.3: Factorization In F [x]
Problem 3E: Find all monic irreducible polynomials of degree 2 over Z3.
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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!
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2Fdacfc428-0c06-4c6e-b2f7-ad095d5806e6%2F1hzl9fe_processed.png&w=3840&q=75)
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.
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