Question

True/False, explain:

1. If G is a finite group and G28, then there is a subgroup of G of
order 2401=7^{4}

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

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 all F-linear transformations of K.

5. Suppose f in F[x] and F in K is a degree 5 field extension. Then Aut(K, F) permutes the roots of f in K.

Answer #1

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. Zx40 is isomorphic to
Zx5 x Zx8
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<S28, then there is
a subgroup of G...

(Modern Algebra) Show that every finite subgroup of the
multiplicative group of a field is cyclical. (Hint: consider m as
the order of the finite subgroup and analyze the roots of the
polynomial (x ^ m) - 1 in field F)

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.

True or False? No reasons needed.
(e) Suppose β and γ are bases of F n and F m, respectively.
Every m × n matrix A is equal to [T] γ β for some linear
transformation T: F n → F m.
(f) Recall that P(R) is the vector space of all polynomials with
coefficients in R. If a linear transformation T: P(R) → P(R) is
one-to-one, then T is also onto.
(g) The vector spaces R 5 and P4(R)...

(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:...

Problem 1. In this problem we work in the finite field
25, i.e. the numbers (mod 5). 1. Show that 2 is a primitive 4-th
root of 1. 2. Show that X1-1= (x - 2)(x - 22)(x - 2)(X – 24). 3.
Show that g(x) = (x - 2)(X - 4) generates a cyclic code C with
d>3. (Hint: invoke a property that we have shown in class.) 4.
What is the generating matrix G of the code C given...

Determine whether the given set ?S is a subspace of the vector
space ?V.
A. ?=?2V=P2, and ?S is the subset of ?2P2
consisting of all polynomials of the form
?(?)=?2+?.p(x)=x2+c.
B. ?=?5(?)V=C5(I), and ?S is the subset of ?V
consisting of those functions satisfying the differential equation
?(5)=0.y(5)=0.
C. ?V is the vector space of all real-valued
functions defined on the interval [?,?][a,b], and ?S is the subset
of ?V consisting of those functions satisfying
?(?)=?(?).f(a)=f(b).
D. ?=?3(?)V=C3(I), and...

Assume that we are working with an aluminum alloy (k = 180
W/moC) triangular fin with a length, L = 5 cm, base thickness, b =
1 cm, a very large width, w = 1 m. The base of the fin is
maintained at a temperature of T0 = 200oC (at the left boundary
node). The fin is losing heat to the surrounding air/medium at T? =
25oC with a heat transfer coefficient of h = 15 W/m2oC. Using the...

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