Question

Let
α = 4√ 3 (∈ R), and

consider the homomorphism

ψα : Q[x] → R

f(x) → f(α).

(a) Prove that irr(α, Q) = x^4 −3

(b) Prove that Ker(ψα) = <x^4 −3>

(c) By applying the Fundamental Homomorphism Theorem to ψα,
prove that

L ={a0+a1α+a2α2+a3α3 | a0, a1, a2, a3 ∈ Q }is the smallest
subfield of R containing α.

Answer #1

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 ·...

Let F be a ﬁeld (for instance R or C), and let P2(F) be the set
of polynomials of degree ≤ 2 with coeﬃcients in F, i.e.,
P2(F) = {a0 + a1x + a2x2 | a0,a1,a2 ∈ F}.
Prove that P2(F) is a vector space over F with sum ⊕ and scalar
multiplication deﬁned as follows:
(a0 + a1x + a2x^2)⊕(b0 + b1x + b2x^2) = (a0 + b0) + (a1 + b1)x +
(a2 + b2)x^2
λ (b0 +...

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....

Let
a1, a2, ..., an be distinct n (≥ 2) integers. Consider the
polynomial
f(x) = (x−a1)(x−a2)···(x−an)−1 in Q[x]
(1) Prove that if then f(x) = g(x)h(x)
for some g(x), h(x) ∈ Z[x],
g(ai) + h(ai) = 0 for all i = 1, 2, ..., n
(2) Prove that f(x) is irreducible over Q

A jar contains three +1, three -1 and four 0. Let Q(x) := a0 +
a1 x + a2 x*x with a0, a1 and a2 drawn from the jar with
replacement.
a). What is the probability that Q has only one root?
b). What is the probability that Q has two real roots?
c). Given that Q has two real roots, what’s the probability that
both of them are positive?

1.- Prove the intermediate value theorem: let (X, τ) be a
connected topological space, f: X - → Y a continuous transformation
and x1, x2 ∈ X with a1 = f (x1), a2 = f (x2) ( a1 different a2).
Then for all c∈ (a1, a2) there is x∈ such that f (x) = c.
2.- Let f: X - → Y be a continuous and suprajective
transformation. Show that if X is connected, then Y too.

Let A = {1, 2, 3, 4, 5}. Describe an equivalence relation R on
the set A that produces the following partition (has the sets of
the partition as its equivalence classes): A1 = {1, 4}, A2 = {2,
5}, A3 = {3} You are free to describe R as a set, as a directed
graph, or as a zero-one matrix.

Consider an axiomatic system that consists of elements in a set
S and a set P of pairings of elements (a, b) that satisfy the
following axioms:
A1 If (a, b) is in P, then (b, a) is not in P.
A2 If (a, b) is in P and (b, c) is in P, then (a, c) is in
P.
Given two models of the system, answer the questions below.
M1: S= {1, 2, 3, 4}, P= {(1, 2), (2,...

Consider the polynomial f(x) = x ^4 + x ^3 + x ^2 + x + 1 with
roots in GF(256). Let b be a root of f(x), i.e., f(b) = 0.
The other roots are b^ 2 , b^4 , b^8 .
e) Write b 4 as a combination of smaller powers of b.
Prove that b 5 = 1. f) Given that b 5 = 1 and the factorization
of 255, determine r such that b = α...

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