Question

True or False, explain:

1. Any polynomial f in Q[x] with deg(f)=3 and no roots in Q is irreducible.

2. Any polynomial f in Q[x] with deg(f)-4 and no roots in Q is irreducible.

3. Z^{x}_{40} is isomorphic to
Z^{x}_{5} x Z^{x}_{8}

4. If G is a finite group and H<G, then [G:H] = |G||H|

5. If [G:H]=2, then H is normal in G.

6. If G is a finite group and G<S_{28}, then there is
a subgroup of G of order 2401=7^{4}

7. If |G|=10, then G is isomorphic to Z_{19}.

8. If F subset of K is a degree 5 field extension, any element b in
K is the root of some polynomial p(x) in F[x]

9. If F subset of K is a degree 5 field extension, viewing K as a vector space over F, Aut(K, F) consists of all F-linear transformationf of K.

Answer #1

True/False, explain:
1. If G is a finite group and G28, then there is a subgroup of G of
order 2401=74
2. If |G|=19, then G is isomorphic to Z19.
3. If F subset of K is a degree 5 field extension, any element b in
K is the root of some polynomial p(x) in F[x]
4. If F subset of K is a degree 5 field extension, viewing K as
a vector space over F, Aut(K, F) consists of...

Let p be an odd prime. Let f(x) ∈ Q(x) be an irreducible
polynomial of degree p whose Galois group is the dihedral group
D_2p of a regular p-gon. Prove that f (x) has either all real roots
or precisely one real root.

Prove that if f(x) =
akx^k
+ak−1x^k+1
+ak−2x^k+2+...+a1x+a0
is a polynomial in Q[x] and ak
̸= 0, and f (x) factors as f
(x) = g(x)h(x),
where g(x) and h(x) are
polynomials in Q[x], then deg f = deg
g+ deg h.

Let f(x) be a cubic polynomial of the form x^3 +ax^2 +bx+c with
real coefficients.
1. Deduce that either f(x) factors in R[x] as the product of
three degree-one
polynomials, or f(x) factors in R[x] as the product of a
degree-one
polynomial and an irreducible degree-two polynomial.
2.Deduce that either f(x) has three real roots (counting
multiplicities) or
f(x) has one real root and two non-real (complex) roots that are
complex
conjugates of each other.

(Sage Exploration) In class, we primarily have worked with the
field Q and its finite extensions. For each p ∈ Z primes, we can
also study the field Z/pZ , which I will also denote Fp, and its
finite extensions. Sage understands this field as GF(p).
(a) Define the polynomial ring S = F2[x].
(b) Find all degree 2 irreducible polynomials. How many are
there? For each,
completely describe the corresponding quadratic field extensions
of F2.
(c) True of false:...

1) find a cubic polynomial with only one root
f(x)=ax^3+bx^2+cx +d such that it had a two cycle using Newton’s
method where N(0)=2 and N(2)=0
2) the function G(x)=x^2+k for k>0 must ha e a two cycle
for Newton’s method (why)? Find the two cycle

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