Question

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)).

Answer #1

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.]

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).

For each of the following pairs of polynomials f(x) and g(x),
write f(x) in the form
f(x) = k(x)g(x) + r(x)
with deg(r(x)) < deg(g(x)).
a) f(x) = x^4 + x^3 + x^2 + x + 1 and g(x) = x^2 −
2x + 1.
b) f(x) = x^3 + x^2 + 1 and g(x) = x^2 − 5x + 6.
c) f(x) = x^22 − 1 and g(x) = x^5 − 1.

Let f(x) g(x) and h(x) be polynomials in R[x].
Show if gcd(f(x), g(x)) = 1 and gcd(f(x), h(x)) = 1, then
gcd(f(x), g(x)h(x)) = 1.

Let f(x) be a nonzero polynomial in F[x]. Show that f(x) is a
unit in F[x] if and only if f(x) is a nonzero constant polynomial,
that is, f(x) =c where 0F is not equal to c where c is a subset of
F. Hence deduce that F[x] is not a field.

Let F be a field. It is a general fact that a finite subgroup G
of (F^*,X) of the multiplicative group of a field must be cyclic.
Give a proof by example in the case when |G| = 100.

1 Approximation of functions by polynomials
Let the function f(x) be given by the following:
f(x) = 1/ 1 + x^2
Use polyfit to approximate f(x) by polynomials of degree k = 2,
4, and 6. Plot the approximating polynomials and f(x) on the same
plot over an appropriate domain. Also, plot the approximation error
for each case. Note that you also will need polyval to evaluate the
approximating polynomial.
Submit your code and both plots. Make sure each of...

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].

Let Z_2 [x] be the ring of all polynomials with coefficients in
Z_2. List the elements of the field Z_2 [x]/〈x^2+x+1〉, and make an
addition and multiplication table for the field. For simplicity,
denote the coset f(x)+〈x^2+x+1〉 by (f(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.)

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