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2
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(1) Let S = 1+y=3 and consider the ring R = Z[o by N(a) = aā = |a|2 =
a² + ab + b2 for a = a +b¢. The norm is multiplicative: N(ab) = N(a)N(B).
(a) What are the units in R?
(b) Show that for every complex number z e C there exists a E R with 2 – al < 1. Use
this to show that R is a Euclidean domain.
(c) Show that for a € R, N(a) is never congruent to 1 modulo 3. Use this to show that if
p is a prime number in Z, then p is also prime as an element of R.
(d) Show that 3 is not prime in R.
(e) Show that if p is a prime congruent to 1 modulo 3, then p is not prime in R. (You may
use the fact that if p is a prime congruent to 1 modulo 3, then there exists an a € Z
such that p divides a² + a +1.
(2) Let w= 1+v-15 and consider the ring R = Z[w] = Z+Zw. Define a positive multiplicative
norm N:R → Zso by N(a) = aā= |a|2 = a² + ab + 462 for a = a + bw.
(a) Show that 1+w is irreducible.
(b) Show that 1+w is not prime.
(c) Give an ideal in R that is not principal.
(3) Suppose that R is an integral domain and R is noetherian (which means that every ideal
in R is finitely generated).
(a) Show that every nonzero element in R that is not a unit can be written as a product
of irreducible elements in R.
(b) Suppose that R is noetherian, and every irreducible element in R is prime. Show that
R is a UFD.
(4) Suppose that R is a UFD.
(a) Suppose that S is a nonzero subring of R with the following property: If a, b e R and
ab E S, then a ES and be S. Prove that S is a UFD as well.
(b) Suppose that T is an integral domain, and ¢ : R → T[x] is a ring homomorphism.
Show that 0-1(T) is a UFD.
(5) Show that the following polynomials are irreducible:
(a) x4 + x3 +1 € F2[x];
(b) x4 + 23x3 – 12x2 + 10x – 37 € Z[x];
(c) x4 + 15x3 + 6x2 – 9x + 2022 € Z[x];
(d) x + 4 € Z[x];
(e) x4 + x²y2 + y2 + y E C[x,y).
(6) Show that the following rings are not noetherian.
(a) The subring R of Z[x] of all polynomials ao + ajx + 2222 + ... + anx" for which
21, 22, ..., An are even (but ao may be odd). (Take the ideal generated by 2x, 2x2, 2x}, ....)
(b) The ring of all functions from and infinite set X to Q. Take the ideal M is in problem 7
of the previous homework set.
=
=
=
=
=
2
=
=
(1) Let S = 1+y=3 and consider the ring R = Z[o by N(a) = aā = |a|2 =
a² + ab + b2 for a = a +b¢. The norm is multiplicative: N(ab) = N(a)N(B).
(a) What are the units in R?
(b) Show that for every complex number z e C there exists a E R with 2 – al < 1. Use
this to show that R is a Euclidean domain.
(c) Show that for a € R, N(a) is never congruent to 1 modulo 3. Use this to show that if
p is a prime number in Z, then p is also prime as an element of R.
(d) Show that 3 is not prime in R.
(e) Show that if p is a prime congruent to 1 modulo 3, then p is not prime in R. (You may
use the fact that if p is a prime congruent to 1 modulo 3, then there exists an a € Z
such that p divides a² + a +1.
(2) Let w= 1+v-15 and consider the ring R = Z[w] = Z+Zw. Define a positive multiplicative
norm N:R → Zso by N(a) = aā= |a|2 = a² + ab + 462 for a = a + bw.
(a) Show that 1+w is irreducible.
(b) Show that 1+w is not prime.
(c) Give an ideal in R that is not principal.
(3) Suppose that R is an integral domain and R is noetherian (which means that every ideal
in R is finitely generated).
(a) Show that every nonzero element in R that is not a unit can be written as a product
of irreducible elements in R.
(b) Suppose that R is noetherian, and every irreducible element in R is prime. Show that
R is a UFD.
(4) Suppose that R is a UFD.
(a) Suppose that S is a nonzero subring of R with the following property: If a, b e R and
ab E S, then a ES and be S. Prove that S is a UFD as well.
(b) Suppose that T is an integral domain, and ¢ : R → T[x] is a ring homomorphism.
Show that 0-1(T) is a UFD.
(5) Show that the following polynomials are irreducible:
(a) x4 + x3 +1 € F2[x];
(b) x4 + 23x3 – 12x2 + 10x – 37 € Z[x];
(c) x4 + 15x3 + 6x2 – 9x + 2022 € Z[x];
(d) x + 4 € Z[x];
(e) x4 + x²y2 + y2 + y E C[x,y).
(6) Show that the following rings are not noetherian.
(a) The subring R of Z[x] of all polynomials ao + ajx + 2222 + ... + anx" for which
21, 22, ..., An are even (but ao may be odd). (Take the ideal generated by 2x, 2x2, 2x}, ....)
(b) The ring of all functions from and infinite set X to Q. Take the ideal M is in problem 7
of the previous homework set.
held.
(7) Suppose that X is a set, and R is the ring of all functions from X to Q. For y e X, let my
be the ideal of all functions f : X → S with f(y) = 0.
(a) Show that my is a maximal ideal for all y E X.
(b) If X is finite, show that every maximal ideal is of the form my.
(c) Suppose X is infinite. Let I be the ideal of all functions f : X + Q for which f(x) = 0
for all but finitely many x E Q. Since I + R there exists a maximal ideal M that
contains I. Prove that M = my for all y.
(d)* Show that the field R/M has uncountably many elements.
E
j
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-
) Let S = 1+
and consider the ring R = Z[5] = Z+Zs. Note that we can view R as
a hexagonal lattice in C. Define a positive norm N: R + Z2o by N(a) = aa = |a|2 =
a2 + ab + b2 for a = a +b5. The norm is multiplicative: N(aß) = N(a)N(B).
(a) What are the units in R?
(b) Show that for every complex number z EC there exists a E R with |z – al < 1. Use
this to show that R is a Euclidean domain.
(c) Show that for a ER, N(a) is never congruent to 2 modulo 3. Use this to show that if
p is a prime number in Z that is congruent to 2 modulo 3, then p is also prime as an
element of R.
(d) Show that 3 is not prime in R.
(e) Show that if p is a prime congruent to 1 modulo 3, then p is not prime in R. (You may
р
use the fact that if p is a prime congruent to 1 modulo 3, then there exists an a E Z
such that p divides a² + a +1.
1+7-15
p
D
7
ՌԸ,
11:
tino
=
=
-
=
-
) Let S = 1+
and consider the ring R = Z[5] = Z+Zs. Note that we can view R as
a hexagonal lattice in C. Define a positive norm N: R + Z2o by N(a) = aa = |a|2 =
a2 + ab + b2 for a = a +b5. The norm is multiplicative: N(aß) = N(a)N(B).
(a) What are the units in R?
(b) Show that for every complex number z EC there exists a E R with |z – al < 1. Use
this to show that R is a Euclidean domain.
(c) Show that for a ER, N(a) is never congruent to 2 modulo 3. Use this to show that if
p is a prime number in Z that is congruent to 2 modulo 3, then p is also prime as an
element of R.
(d) Show that 3 is not prime in R.
(e) Show that if p is a prime congruent to 1 modulo 3, then p is not prime in R. (You may
р
use the fact that if p is a prime congruent to 1 modulo 3, then there exists an a E Z
such that p divides a² + a +1.
1+7-15
p
D
7
ՌԸ,
11:
tino
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