Abstract algebra – Why does the integral domain "enclosed between a finite field extension" implies that it is a field?

The following is an exercise by Qing Liu Algebraic geometry and arithmetic curves,

Exercise 1.2.

Let φ: A → B be a homomorphism of the finitely generated algebras over a field. Show that the image of a closed point under Spec φ is a closed point.

The following is the solution of Cihan Bahran. http://www-users.math.umn.edu/~bahra004/alg-geo/liu-soln.pdf.

Write $ k $ for the field below. Let us analyze the statement. A closed point in $ operatorname {Spec} B $ means a maximum ideal $ n $ from $ B $, And $ operatorname {Spec} (φ) (n) = φ ^ {- 1} (n) $, So let's show that $ p: = φ {-1} (n) $ is a maximum ideal in $ A $, First, $ p $ is definitely an ideal ideal of $ A $ and $ φ $ descends to an injectable $ k $Algebra homomorphism $ ψ: A / p to B / n $, But the card $ k to B / n $ Defines a finite field extension of $ k $ from Corollary 1.12. So the holistic domain $ A / p $ is included between a finite field extension. Such domains are necessarily fields $ p $ is maximum in $ A $,

In the penultimate sentence, the author says that the integral area $ A / p $ is included between a finite field extension. I do not know exactly what it means, but I think there are two injective ring homomorphisms $ f: k to A / p $ and $ g: A / p to B / n $ so that $ g circ f $ makes $ B / n $ a finite field extension of $ k $, But why is that? $ A / p $ is a field?

Algebra precalculus – Alegbaraic terms of higher order

I have learned from Stroud & Booth's amazing "Engineering Mathematics" myself and am currently in the field of "Algebra".

I have been working on the distribution of algebraic expressions, and the book explains how to solve the simpler ones, such as:

$$ (2 {y} ^ 2 -y-10) div (y + 2) $$

But for the practice exercises I have one of which I have absolutely no idea where to start, and none of the examples or problems that have been in the book so far come close to this:

$$ frac {(2r ^ 3 + 5r ^ 2 – 4r + 3} {r ^ 2 + 2r – 3} $$

Can someone help me or please point me in the right direction?

Linear Algebra – Need help finding matrix eigenvectors

I have the following matrix:

$$ B = begin {bmatrix}
8 & -6 & -6 \
30 & -22 & -30 \
-30 & 30 & 38
end {bmatrix} $$

I find the characteristic polynomial as:

$$ det (B- lambda E) = – ( lambda-8) ^ 3 $$

So we have the root 8 (with multiplication 3). That's why $ B $ has the eigenvalue 8

Now I dissolve

$$ (B- Lambda E) x = begin {bmatrix}
8-8 & -6 & -6 \
30 & -22-8 & -30 \
-30 & 30 & 38-8
end {bmatrix}
begin {bmatrix}
x \
y \
z
end {bmatrix} =
begin {bmatrix}
0 \
0 \
0
end {bmatrix} $$

$$ 0x-6y-6z = 0 $$
30x-30y-30z = 0 $$
$$ – 30x + 30y + 30z = 0 $$

I find the solution to the following equations as:

$$ x = 0 $$
$$ y = -1 $$
$$ z = 1 $$

Am I in the right direction? What do the results tell me about which eigenvectors exist for matrix B?

Algebra precalculus – Importance of vertical asymptote for a particular problem

Problem: Suppose a rocket is fired upwards from the Earth's surface at an initial velocity v (measured in meters per second). Then the maximum height h (in meters) reached by the rocket is given by the function h (v) = Rv22gR – v2, where R = 6.4 × 106 m the radius of the earth and g = 9.8 m / s2 the due acceleration is to gravity. Draw a graphic of the function h with a graphics device. (Note that h and v must both be positive, so the display rectangle does not need to contain negative values.)

What does the vertical asymptote physically mean?

Linear algebra – Changing scalar products in a vector space different dimensions?

I am looking for a abelsche group $ (G, +) $ and a field$ (F, +, ×) $ and two scalar products $ ._ 1 $ and $ ._ 2 $ such that vector spaces are caused by first and second scalar products $ G $ over $ F $have different dimensions.

I am interested in an example with an infinite field, but first any example would be useful (if any, if there is no such example, it would be great to give a proof).

Linear Algebra – How are "holes" filled in a vector? [1 0 1] -> [1 1 1]

Support I have a vector that looks like this:
[0 0 1 0 1 0 0 1 1]
Which algebraic operation fills the "holes" between 2 "1" and makes it a field of "1"?

What I mean is how can I transform this vector into:
[0 0 1 1 1 0 0 1 1]
(The fourth point was changed from "0" to "1" because it was a 0 between 2 "1")

A kind of folding?

linear algebra – Do you determine all equivalence classes of a given quadratic form?

To let $ G (x, y) $ be a binary quadratic form of an integral coefficient that is positively determined.
If the discriminant of $ G (x, y) $ is -56 $then I have to determine all equivalence classes of $ G (x, y) $,

Say $ G (x, y) = x ^ {t} Ax $, from where $ | A | = left | begin {array} {cccc} a & b \ c & d end {array} right | $then I can determine that $ ad-bc = -56 $ and $ a> 0 $, in addition, $ G (x, y) $ corresponds to a reduced form, where $ 0 <a leq 4 $ and $ 0 leq | b | leq 2 $,

Do I have to use this conclusion? Or how can I determine all? Please help.

ac.commutative algebra – Are the ring of the power series and the seed ring of the holomorphic functions catenary?

Yes, they are regular (the maximum ideal is generated by a number of elements corresponding to their dimension), and therefore Cohen-Macaulay (Matsumura, Theorem 17.8). A Cohen-Macaulay ring is a chain line (Matsumura, Theorem 17.4).

Matsumura, Commutative Ring Theory, CUP, 1986

Abstract algebra – Which fields between the rational and the real allow a good idea of ​​2D distance?

Consider a field $ K $, we say $ K subseteq mathbb R $, We can look at the "plane" $ K times K $, I wonder in which cases the distance function works $ d: K times K to mathbb R $, defined as normal by $ d (x, y) = sqrt {x ^ 2 + y ^ 2} $takes values ​​in $ K $,

Certainly this is not true $ mathbb Q $: we have $ d (1, 1) = sqrt {2} notin mathbb Q $, If we take some $ K $ which is closed under square roots, then certainly $ d $ will accept values $ K $,

It may still be true a priori $ a in K $ has no square root, but this is not an obstacle, as there is no spelling $ a = x ^ 2 + y ^ 2 $, So I am surprised:

Are there fields? $ K subseteq mathbb R $ which do not have all square roots, but are still closed $ d $?

linear algebra – Show that the function $ || u || = sqrt {2 | u_1 | ^ 2 + 5 | u_2 | ^ 2} $ is a norm for $ V $.

I am given: let $ V $ be the real vector space $ mathbb {R} ^ 2 $, and $ u = [u_1 , u_2]^ T in V $, Show that the function $ || u || = sqrt {2 | u_1 | ^ 2 + 5 | u_2 | ^ 2} $ is a norm on $ V $, Then determine if this norm is derived from an inner product. If so, what is the inner product?

After many attempts, I have difficulty proving (a) that this function satisfies the property of the triangular inequality of norms (i $ || u + v || leq || u || + || v || $) and (b) determining the internal product from which this standard is derived. Any help that comes off, would be very grateful!