Question

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

Answer #1

Let S = {(a1, a2, . . . , an)| n ≥ 1, ai ∈ Z ≥0 for i = 1, 2, .
. . , n, an 6= 0}. So S is the set of all finite ordered n-tuples
of nonnegative integers where the last coordinate is not 0. Find a
bijection from S to Z +.

10. Let P(k) be the following statement: ”Let a1, a2, . . . , ak
be integers and p be a prime. If p|(a1 · a2 · a3 · · · ak), then
p|ai for some i with 1 ≤ i ≤ k.” Prove that P(k) holds for all
positive integers k

Let S = {(a1,a2,...,an)|n ≥ 1,ai ∈ Z≥0 for i = 1,2,...,n,an ̸=
0}. So S is the set of all finite ordered n-tuples of nonnegative
integers where the last coordinate is not 0. Find a bijection from
S to Z+.

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 ∈ Z[x] be a nonconstant polynomial with the property that
all the roots (in comlex plane) for the equation f(x) = 0 are
distinct. Prove that there exist infinitely many positive integers
n such that f(n) is not a perfect square.

Let f ∈ Z[x] be a nonconstant polynomial with the property that
all the roots (in comlex plane) for the equation f(x) = 0 are
distinct. Prove that there exist infinitely many positive integers
n such that f(n) is not a perfect square. Could you explain it in
number theory instead of some deep math like sigel theorem

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....

1.13. Let a1, a2, . . . , ak be integers with gcd(a1, a2, . . .
, ak) = 1, i.e., the largest
positive integer dividing all of a1, . . . , ak is 1. Prove that
the equation
a1u1 + a2u2 + · · · + akuk = 1
has a solution in integers u1, u2, . . . , uk. (Hint. Repeatedly
apply the extended Euclidean
algorithm, Theorem 1.11. You may find it easier to prove...

Suppose V is a vector space over F, dim V = n, let T be a linear
transformation on V.
1. If T has an irreducible characterisctic polynomial over F,
prove that {0} and V are the only T-invariant subspaces of V.
2. If the characteristic polynomial of T = g(t) h(t) for some
polynomials g(t) and h(t) of degree < n , prove that V has a
T-invariant subspace W such that 0 < dim W < n

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