Bezout’s Theorem and the Fundamental Theorem of Arithmetic
1. Let a, b, c ∈ Z. Prove that c = ma + nb for some m, n ∈ Z if
and only if gcd(a, b)|c.
2. Prove that if c|ab and gcd(a, c) = 1, then c|b.
3. Prove that for all a, b ∈ Z not both zero, gcd(a, b) = 1 if
and only if a and b have no prime factors in common.
According to the Fundamental Theorem of Algebra, every
nonconstant polynomial f (x) ∈
C[x] with complex coefficients has a complex root.
(a) Prove every nonconstant polynomial with complex coefficients
is a product of linear polynomials.
(b) Use the result of the previous exercise to prove every
nonconstant polynomial with real coefficients is a product of
linear and quadratic polynomials with real coefficients.
Prove by induction that it follows from Fundamental Theorem of Algebra that every f(x) ∈ C[x] can be written into a product of linear polynomials in C[x].
Verify the Divergence Theorem for the vector field F(x, y, z) =
< y, x , z^2 > on the region E bounded by the planes y + z =
2, z = 0 and the cylinder x^2 + y^2 = 1.
By Surface Integral:
By Triple Integral:
Use the extended divergence theorem to compute the total flux of
the vector field
F(x, y, z) = −3x2 + 3xz − y, 2y3 − 6y, 9x2 + 4z2 − 3x outward
from the region F that lies inside the sphere x2 + y2 + z2 = 25 and
outside the solid cylinder x2 + y2 = 4 with top at z = 1 and bottom
at z = −1.
Theorem: Let K/F be a field extension and let a ∈ K be algebraic
over F. If deg(mF,a(x)) = n, then
1. F[a] = F(a).
2. [F(a) : F] = n, and
3. {1, a, a2 , ..., an−1} is a basis for F(a).
Verify that the Divergence Theorem is true for the vector field
F on the region E. Give the flux. F(x, y, z) = xyi + yzj + zxk, E
is the solid cylinder x2 + y2 ≤ 144, 0 ≤ z ≤ 4.
Prove the theorem in the lecture:Euclidean Domains and UFD's
Let F be a field, and let p(x) in F[x]. Prove that (p(x)) is a
maximal ideal in F[x] if and only if p(x) is irreducible over
F.
(abstract algebra) Let F be a field. Suppose f(x), g(x), h(x) ∈
F[x]. Show that the following properties hold:
(a) If g(x)|f(x) and h(x)|g(x), then h(x)|f(x).
(b) If g(x)|f(x), then g(x)h(x)|f(x)h(x).
(c) If h(x)|f(x) and h(x)|g(x), then h(x)|f(x) ± g(x).
(d) If g(x)|f(x) and f(x)|g(x), then f(x) = kg(x) for some k ∈ F
\ {0}