Let 
be a field of characteristic 
and 
in ![F[x]](http://img.homeworklib.com/questions/a9667a20-ab49-11ea-8fd1-af9da2fcbd0b.png?x-oss-process=image/resize,w_560)
.
i.) Suppose 
has a zero 
in
. Show
splits in
and find the factorization of
ii.)Suppose
does not have a zero in
. Let
be a zero of
in an extension of
. Show
splits in 
and find a factorization of
.
Homework Answers
![# -C (14 Given that if beg field of charecteristic p. let for = xp -q Elf [x] det f(x) = xch_a has a zero bEIF. Then focb) =](http://img.homeworklib.com/questions/74a2ae20-240a-11eb-b456-7d0b4ba0da2f.png?x-oss-process=image/resize,w_560)
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Let
be a field of characteristic
and
in
.
i.) Suppose
has a zero
in
....
Let be an arbitrary function and A X. i) Show that A ii) Give an example...
Let
be an arbitrary function and A
X.
i) Show that A
ii) Give an example to show that in general A =
.
iii) Show that, if
is injective, then A =
iv) Show that, if X and Y are modules;
is a homomorphism of modules and A is a submodule of X such that
ker,
then we also have A =
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Let be i.i.d. . Define the sample mean and the sample variance by and . (i)...
Let be i.i.d. . Define the sample mean and the sample variance by and . (i) Find the distribution of and for i = 1, ... , n. (ii) Show that and are independent for i = 1, ... , n. (iii) Hence, or otherwise, show that and are independent. 7l N (μ, σ2) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were...
Let U ⊆ R^n be open (not necessarily bounded), let f, g : U → R...
Let U ⊆ R^n be open (not necessarily bounded), let f, g : U → R
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that if
exists, then so does
.
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Let be a prime and let be the set of rational numbers whose denominator (when written...
Let be a prime and let be the set of rational numbers whose denominator (when written in lowest terms) is not divisible by . i) Show, with the usual operations of addition and multiplication, that is a subring of . ii) Show that is a subring of . iii) Is a field? Explain. iv) What is where is the set of all fractions with denominator a power of We were unable to transcribe this imageWe were unable to transcribe this...
Let X1, X2, ..., Xn be a random sample from X which has pdf depending on...
Let X1, X2, ..., Xn be a random
sample from X which has pdf
depending on a parameter
and
(i)
(ii)
where
< x <
. In both these two cases
a) write down the log-likelihood function and find a
1-dimensional sufficient statistic for
b) find the score function and the maximum likelihood estimator
of
c) find the observed information and evaluate the Fisher
information at
= 1.
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Abstract Algebra: Let . It has been shown already that K is the splitting field over...
Abstract Algebra: Let
. It has been shown already that K is the splitting field over
, and the
following isomorphisms are of onto a subfield
as extensions of the automorphism
, and also the elements of :
;
;
;
.
We also proved previously that is separable over
. Based
on all of those outcomes, find all subgroups of
and their corresponding fixed fields as the intermediate fields
between and
, and
complete the subgroup and subfield diagrams...
For any vector field F⃗ and any scalar function f we define a new field a) Assuming that the app...
For any vector field F⃗ and any scalar function f we define a new field
a) Assuming that the appropriate partial derivatives are
continuous, show the following formula:
b) Let ⃗x = x⃗i + y ⃗j + z ⃗k and the vector field
Use the formula found in a) to answer
the following question: is there a number p such that F⃗ is incompressible (that is, its divergence is zero)?
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3. Let ,..., be independent random sample from N(), where is unknown. (i) Find a sufficient...
3. Let ,..., be
independent random sample from N(),
where is unknown.
(i) Find a sufficient statistic of .
(ii) Find the MLE of .
(iii) Find a pivotal quantity and use it to construct a
100(1–)% confidence
interval for .
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Let , ... be independent random variables with mean zero and finite variance. Show that
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Let ⊂ be a rectangle and let f be a function which is integrable on R....
Let ⊂
be a
rectangle and let f be a function which is integrable on R. Prove
that the graph of f, G(f) := {(x, f(x)) ∈
: x ∈ }, is a
Jordan region and that it has volume 0 (as a subset of
).
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let
be an arbitrary function and A
X.
i) Show that A
ii) Give an example to show that in general A =
.
iii) Show that, if
is injective, then A =
iv) Show that, if X and Y are modules;
is a homomorphism of modules and A is a submodule of X such that
ker,
then we also have A =
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe...
Let U ⊆ R^n be open (not necessarily bounded), let f, g : U → R
be continuous, and suppose that |f(x)| ≤ g(x) for all x ∈ U. Show
that if
exists, then so does
.
We were unable to transcribe this imageWe were unable to transcribe this image
Let X1, X2, ..., Xn be a random
sample from X which has pdf
depending on a parameter
and
(i)
(ii)
where
< x <
. In both these two cases
a) write down the log-likelihood function and find a
1-dimensional sufficient statistic for
b) find the score function and the maximum likelihood estimator
of
c) find the observed information and evaluate the Fisher
information at
= 1.
f(20) We were unable to transcribe this image((z(0 – 2) - )dxəz(47)...
Abstract Algebra: Let
. It has been shown already that K is the splitting field over
, and the
following isomorphisms are of onto a subfield
as extensions of the automorphism
, and also the elements of :
;
;
;
.
We also proved previously that is separable over
. Based
on all of those outcomes, find all subgroups of
and their corresponding fixed fields as the intermediate fields
between and
, and
complete the subgroup and subfield diagrams...
For any vector field F⃗ and any scalar function f we define a new field
a) Assuming that the appropriate partial derivatives are
continuous, show the following formula:
b) Let ⃗x = x⃗i + y ⃗j + z ⃗k and the vector field
Use the formula found in a) to answer
the following question: is there a number p such that F⃗ is incompressible (that is, its divergence is zero)?
f F)(x,y,z) = f(x,y,z)F(x,y, z) We were unable to transcribe...
3. Let ,..., be
independent random sample from N(),
where is unknown.
(i) Find a sufficient statistic of .
(ii) Find the MLE of .
(iii) Find a pivotal quantity and use it to construct a
100(1–)% confidence
interval for .
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable...
Let ⊂
be a
rectangle and let f be a function which is integrable on R. Prove
that the graph of f, G(f) := {(x, f(x)) ∈
: x ∈ }, is a
Jordan region and that it has volume 0 (as a subset of
).
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
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