Question

problem 2 In the polynomial ring Z[x], let I = {a0 + a1x + ... +
anx^n: ai in Z[x],a0 = 5n}, that is, the set of all polynomials
where the constant coefficient is a multiple of 5. You can assume
that I is an ideal of Z[x]. a. What is the simplest form of an
element in the quotient ring z[x] / I? b. Explicitly give the
elements in Z[x] / I. c. Prove that I is **not a principal
ideal** (that is, I does not equal <f(x)> for any
polynomial f(x) in Z[x])

Answer #1

Prove that if f(x) =
akx^k
+ak−1x^k+1
+ak−2x^k+2+...+a1x+a0
is a polynomial in Q[x] and ak
̸= 0, and f (x) factors as f
(x) = g(x)h(x),
where g(x) and h(x) are
polynomials in Q[x], then deg f = deg
g+ deg h.

Let p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2 be vectors
in P2 with p, q = a0b0 + a1b1 + a2b2. Determine whether the
polynomials form an orthonormal set, and if not, apply the
Gram-Schmidt orthonormalization process to form an orthonormal set.
(If the set is orthonormal, enter ORTHONORMAL in both answer
blanks.)
{−2 + x2, −2 + x}
u1=
u2=

Let p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2 be vectors
in P2 with p, q = a0b0 + a1b1 + a2b2. Determine whether the given
second-degree polynomials form an orthonormal set, and if not, then
apply the Gram-Schmidt orthonormalization process to form an
orthonormal set. (If the set is orthonormal, enter ORTHONORMAL in
both answer blanks.) { 3 (x2−1), 3 (x2 + x + 2)}
u1 =
u2 =

Let p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2 be vectors
in P2 with p, q = a0b0 + a1b1 + a2b2. Determine whether the given
second-degree polynomials form an orthonormal set, and if not, then
apply the Gram-Schmidt orthonormalization process to form an
orthonormal set. (If the set is orthonormal, enter ORTHONORMAL in
both answer blanks.)
{ square root 3 (x2−1), square root 3 (x2 + x + 2)}
u1 =
u2...

Let I= (x2 +2) in Z7 [x] , and let
R be the factor ring Z7 [x] / I.
a) Prove that every element of R can be written in the
form f + I where f is an element of Z7 [x] and
deg(f0< or =2 or f=0. That is,
R={ f + I : f in Z7 [x] and (deg (f) , or=2
or f=0)}

Let R be a commutative ring with unity. Prove that the principal
ideal generated by x in the polynomial ring R[x] is a prime ideal
iff R is an integral domain.

For each polynomial f(x) ∈ Z[x], let f ' (x) denote its
derivative, which is also a polynomial in Z[x]. Let R be the
following subset of Z[x]: R = {f(x) ∈ Z[x] | f ' (0) = 0}. (a)
Prove that R is a subring of Z[x]. (b) Prove that R is not an ideal
of Z[x].

4. (30) Let C be the ring of complex numbers,and letf:C→C be the
map defined by
f(z) = z^3.
(i) Prove that f is not a homomorphism of rings, by finding an
explicit counterex-
ample.
(ii) Prove that f is not injective.
(iii) Prove that the principal ideal I = 〈x^2 + x + 1〉 is not a
prime ideal of C[x].
(iv) Determine whether or not the ring C[x]/I is a field.

Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It turns out that R forms a ring under the operations:
(a1,a2,a3,···)+(b1,b2,b3,···)=(a1 +b1,a2 +b2,a3 +b3,···),
(a1,a2,a3,···)·(b1,b2,b3,···)=(a1 ·b1,a2 ·b2,a3 ·b3,···)
Let I = {(a1,a2,a3,···) ∈ Z∞ : all but finitely many ai are 0}.
You may use without proof the fact that I forms an ideal of R.
a) Is I principal in R? Prove your claim.
b) Is I prime in R? Prove your claim....

Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It turns out that R forms a ring under the operations:
(a1,a2,a3,···)+(b1,b2,b3,···)=(a1 +b1,a2 +b2,a3 +b3,···),
(a1,a2,a3,···)·(b1,b2,b3,···)=(a1 ·b1,a2 ·b2,a3 ·b3,···)
Let I = {(a1,a2,a3,···) ∈ Z∞ : all but finitely many ai are 0}.
You may use without proof the fact that I forms an ideal of R.
a) Is I principal in R? Prove your claim.
b) Is I prime in R? Prove your claim....

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