Question

Let E/F be an algebraic extension. Let K and L be intermediate fields (i.e. F ⊆ K ⊆ E and F ⊆ L ⊆ E).

(i) Prove that if the extension K/F is separable then the extension KL/L is separable.

(ii) Prove that if the extension K/F is normal then the extension KL/L is normal.

Note: To make things easier for you, you can assume that E/F is finite (hence all extensions are finite),

Answer #1

Let F⊆K⊆E be extension fields. If K is an algebraic extension of
F and let α∈E be algebraic over K. Show that α is also algebraic
over F.

Let E/F be a finite Galois extension such that Gal(E/F) is
abelian. Prove that for
every intermediate field K, the extension K/F is Galois.

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 the following represents the economy
Y = F (K, L) = √ KL
s = 0.3
δ = 0.1
k0 = 4
Suppose that the steady-state occurs in the year “SS” above.
Calculate the steady-state level of capital per worker, and fill in
the remaining rows, (SS) to (SS+2).
What do you notice happens to economic growth, gy, once steady
state is reached (and beyond the steady state)? Why does this
happen?
Note: Economic growth is
gY = Yt...

Fluoride ion selective electrodes obey the following
equation:
E(V) = K -
β(0.0592)log[F-]
where the fluoride concentration is expressed as mol/L and the
temperature coefficient β is normally equal to 1.
In a normal calibration procedure you dilute a standard fluoride
solution to between 10 and 20 mg/L fluoride ion and then measure
the response of the electrode. You then plot the data and do a
linear regression to obtain the fitting coefficients K and β. Use
the plotting and...

Use python language please
#One of the early common methods for encrypting text was
the
#Playfair cipher. You can read more about the Playfair cipher
#here: https://en.wikipedia.org/wiki/Playfair_cipher
#
#The Playfair cipher starts with a 5x5 matrix of letters,
#such as this one:
#
# D A V I O
# Y N E R B
# C F G H K
# L M P Q S
# T U W X Z
#
#To fit the 26-letter alphabet into...

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...

As you saw from the lab PowerPoint slides last week, you will be
doing a research study looking at ‘Aggression Priming” for your
first paper. For this week’s discussion, I want you to discuss with
your group what you think this study is about. What is the
hypothesis? What theory does it come from? What do you predict will
happen (do you expect something different than the hypothesis in
the researcher instructions? If so, what and why?)? Do you think...

Do the following problems. 1. Each of three barrels from a
manufacturing line are classified as either above (a) or below (b)
the target weight. Provide the ordered sample space. 2. The heat on
each of two soldered parts is measured and labeled as either low
(l), medium (m), or high (h). State the number of elements in the
ordered sample space. 3. Consider the set of Beatles songs with a
primary writer as either Paul McCartney (P) or John...

Two computer users were discussing tablet computers. A higher
proportion of people ages 16 to 29 use tablets than the proportion
of people age 30 and older. The table below details the number of
tablet owners for each age group. Test at the 1% level of
significance. (For subscripts let 1 = 16-29 year old users, and 2 =
30 years old and older users.)
16–29 year
olds
30 years old and
older
Own a
Tablet
69
231
Sample
Size...

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