Question

Let F be a field and f(x), g(x) ? F[x] both be of degree ? n. Suppose that there are distinct elements c0, c1, c2, · · · , cn ? F such that f(ci) = g(ci) for each i. Prove that f(x) = g(x) in F[x].

Answer #1

**Proof:**

Let _{
} . Suppose _{}
then _{
} .

Let _{}
be elements in
such that _{
}.

Then _{}

_{}
are zeros of .

Hence **, **
has _{}
zeros which is not possible.

_{}
must equal .

Hence **,** _{
}.

Let F be a field and let f(x) be
an element of F[x] be an an irreducible
polynomial. Suppose K is an extension field containing F and that
alpha is a root of f(x). Define a function f: F[x] ---> K by
f:g(x) = g(alpha). Prove the ker(f) =<f(x)>.

1. The Taylor series for f(x)=x^3 at 1 is ∞∑n=0 cn(x−1)^n.
Find the first few coefficients.
c0=
c1=
c2=
c3=
c4=
2. Given the series:
∞∑k=0 (−1/6)^k
does this series converge or diverge?
diverges
converges
If the series converges, find the sum of the series:
∞∑k=0 (−1/6)^k=

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

Let
F be a field and let a(x), b(x) be polynomials in F[x]. Let S be
the set of all linear combinations of a(x) and b(x). Let d(x) be
the monic polynomial of smallest degree in S. Prove that d(x)
divides a(x).

Let
F be a field, and let f (x), g(x) ∈ F [x] be nonzero polynomials.
Then it must be the case that deg(f (x)g(x)) = deg(f (x)) +
deg(g(x)).

Let
E/F be a field extension, and let α be an element of E that is
algebraic over F.
Let p(x) = irr(α, F) and n = deg p(x).
(a) For f(x) ∈ F[x], let r(x) (∈ F[x]) be the remainder of
f(x) when divided by p(x).
Prove that f(x) +p(x)= r(x)+p(x)in F[x]/p(x).
(b) Prove that if |F| < ∞, then | F[x]/p(x)| = |F|n. (For a
set A, we denote by |A| the number of elements in A.)

Suppose n = rs where r and s are distinct primes, and let p be a
prime. Determine (with proof, of course) the number of irreducible
degree n monic polynomials in Fp[x]. (Hint: look at the proof for
the number of prime degree polynomials)
The notation Fp means the finite field with q elements

(a) Let a,b,c be elements of a field F. Prove that if a not= 0,
then the equation ax+b=c has a unique solution.
(b) If R is a commutative ring and x1,x2,...,xn are independent
variables over R, prove that R[x σ(1),x σ (2),...,x σ (n)] is
isomorphic to R[x1,x2,...,xn] for any permutation σ of the set
{1,2,...,n}

(a) Suppose f and g are di?erentiable on an interval I and that
f(x) − ((g(x))n = c for all
xεI (where nεNand cεR are constants). If g(x) ̸= 0 on I, then
g′(x) = −f(x)((g(x))1−n.n
(b) If f is not di?erentiable at x0,then f is not continuous at
x0.
(c) Suppose f and g are di?erentiable on an interval I and
suppose that f′(x) = g′(x)on I.
Then f(x) = g(x) on I.
(d) The equation of the line...

Let f(x) and g(x) be polynomials and suppose that we have f(a) =
g(a) for all real numbers a. In this case prove that f(x) and g(x)
have exactly the same coefficients. [Hint: Consider the polynomial
h(x) = f(x) − g(x). If h(x) has at least one nonzero coefficient
then the equation h(x) = 0 has finitely many solutions.]

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