This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

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0
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3answers
72 views

Just a proof of algebra

If $a+b+c=0$, Show that $\left[\dfrac{a}{b-c}+\dfrac{b}{c-a}+\dfrac{c}{a-b}\right]\left[\dfrac{b-c}{a}+\dfrac{c-a}{b}+\dfrac{a-b}{c}\right]=9.$ I am struck with this problem but can't find a ...
2
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2answers
35 views

In Zagier's one-sentence proof, why is S defined to be {(x,y,z)∈ℕ^3:x^2+4yz=p,p prime}?

I've looked at a very clear explanation of Zagier's proof (specifically, it can be found here:http://danielmath.wordpress.com/2012/12/26/one-sentence-proof/) but the first step still eludes me: why is ...
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1answer
48 views

Localization of Rings: Show $R_{f} \simeq R[X]/\langle 1-fX \rangle$

Let $R$ be a ring, $f \in R$, and $X$ a variable. Show $R_{f} \simeq R[X]/\langle 1-fX \rangle$. I am a beginner in algebra and I am reading a textbook in commutative algebra. What I do not ...
0
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1answer
45 views

Explanation of Proof that field sum of more than 2 elements is 0. [duplicate]

"Suppose the field $F$ is finite. If $f\colon F\to F$ is any bijection, then we can conclude that $\sum_{x\in F}x=\sum_{x\in F}f(x)$. Let $\alpha\in F$ such that $\alpha\ne 0$. Then $x\mapsto \alpha ...
-2
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0answers
9 views

Why doesnt a (43, 43, 7, 7, 1)-design exist according to the conditions?

I have tested this using the necessary conditions for a BIBD and it's giving me a green light but I know this isn't a design. Why not?
2
votes
0answers
43 views

Is proper morphism from affine scheme affine?

I'm reading Mumford-Oda's lecture notes http://www.math.upenn.edu/~chai/624_08/mumford-oda_chap1-6.pdf. And they use the fact:"Let $f:U \to Y$ be a proper morphism of noetherian schemes and $U$ is ...
2
votes
2answers
49 views

Prove that the unity element in a subfield of a field must be the unity of the whole field

The solution I was given says: Let $F$ be a field and suppose $u^2=u$ for some nonzero $u$ in $F$. By multiplying each side by $u^{-1}$ it is clear that $0$ and $1$ are the only solutions of ...
1
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1answer
27 views

Explaining That there is no nontrivial ring homomorphism between Z and nZ

My instructor wrote in his notes the following example: "As groups (Z,+) and (nZ,+) are isomorphic. As rings is there any nontrivial homomorphism $\phi$: Z->nZ? The answer is no and he gives the ...
5
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1answer
84 views

Construction of a triple cover of $A_6$ in “Finite Simple Groups” by Wilson

I am reading The Finite Simple Groups by Robert Wilson: see page 29. I want to understand a construction of triple cover of $A_6$. On section 2.7.3., I don't understand the second paragraph, which is ...
5
votes
1answer
64 views

Satellite functors in Cartan Eilenberg

I was reading and came across this statement whose proof is said to be obvious. I however after hours still cannot figure out how to prove $S_2T(A) = S_1(S_1T(A)) = S_1T(M)$. The definitions are: ...
3
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1answer
60 views

Doubt on a paragraph regarding Lagrange's multiplier.

I've a topic in my notes "The method of Lagrange's multipliers" which is described as follows: Let $U$ be an open set in $\mathbb R^n$.Let $f\in C^1(U,\mathbb R)$ and let ...
1
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1answer
34 views

Question on abstract algebra about Group?

I need an explanation, why $ (\mathbb{Z}_7,\oplus _6 )$ is not a Group? As I have discovered so far. The following conditions are satisfied I) Closed! II) Associative! III) ...
0
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4answers
40 views

why is $\sum_0^{n-1} k(k-1) = n(n-1)(n-2)/3$ [closed]

how do I get that $\sum_0^{n-1} k(k-1) = n(n-1)(n-2)/3$? And why is it true?
0
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1answer
17 views

Sequence Lemma explanation

Then every neighbourhood $U$ of $x$ contains a point of $A$. So I don't see it happening unless $X$ is a metric space, but the proof is for any topological space.
0
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1answer
30 views

Haudorff Formula Set Theory

For every $\alpha$ and every $\beta$, $$\aleph_{\alpha+1}^{\aleph_{\beta}}=\aleph_{\alpha}^{\aleph_{\beta}} \cdot \aleph_{\alpha+1}$$ Proof: If $\beta \geq \alpha+1$, then ...
0
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0answers
20 views

Fichtenholz's limit proof for a sequence

I'm reading calculus written by G.M Fichtenholz. And i can't understand his proof. The proof is: $n>2$ There is a sequence $x_n = \frac{n^2 -n + 2}{3n^2+2n-4}$ $x_n - \frac13 = \frac{5n - 10}{3( ...
1
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1answer
21 views

Question regarding an algebraic manipulation in GFology

How does the author arrive at the last equality in the first line, i.e.$$\text{why is } [x^k]\frac{1}{1-y(1+x)} = \frac{1}{1-y}[x^k]\frac{1}{1-\left(\frac{y}{1-y}\right)x} \text{?}$$
1
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2answers
56 views

set theory proof explanation

The following lemma is taken from the book 'Introduction to Set Theory' by Hrbacek and Jech. chapter $6$ normal form Can anyone explain to me why the first sentence holds ( the existence of ...
0
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0answers
24 views

Multilinear Function proof in Spivak?

Note that $$\wedge^n (V)$$ denotes the set of all alternating multilinear functions and $\mathfrak{I}^n(R^n)$ denotes the set of all multilinear function. I don't know what the actual symbol ...
2
votes
1answer
30 views

Multilinear algebra and matrices

Given $\wedge^k(V)$ an alternating multilinear space and $T : V \to W$ a linear map, then we have $$v_1 \wedge \dots \wedge v_k \in\wedge^k(V).$$ Define $$\wedge^k(T)(v_1\wedge\dots\wedge v_k) = ...
0
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1answer
38 views

Understanding the proof that $c_0$ is a closed subspace of $\ell^\infty$

The problem is given: source Let $c_0$ be a space of real sequences $x = \{x_n\}_{n = 1}^\infty=0$ converging to $0$. Let $\ell^\infty$ be a set of real sequences $w = \{w_k\}^\infty_{k=0}$ ...
0
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1answer
74 views

A limit theorem in Rudin. Please elaborate?

Theorem: Let $\{ p_n \} \in X $. If $E \subset X$ and $p$ is a limit point of $E$, then there is a sequence $\{ p_n \}$ such that $p = \lim p_n $ The proof goes like this proof: For each ...
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0answers
37 views

Transfinite Induction (Proof Explanation)

The theorem above is extracted from the book 'Introduction to Set Theory' by Hrbacek and Jech. Questions: $1$)I don't understand the successor case. When $\alpha_2=\beta+1$, why suddenly $W_2$ ...
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1answer
35 views

Help understanding example in Engel's *Problem Solving Strategies*

I've spent a lot of time trying to follow the chain of reasoning, but to no avail. I lose track of how it works at the "Adding (1) and (2)" part. Could someone help me understand this, please?
1
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1answer
46 views

Proposition 0.16 in Hatcher's AT

In the proof of the quoted proposition, it is mentioned that $D^n \times I$ retracts onto $D^n \times \left\{0\right\} \cup \partial D^n \times I$ and an example is given in a figure with $n=2$, which ...
0
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1answer
59 views

$1$ is not congruent because of Fermat's Last Theorem?

I would like someone to explain something I did not understand. I was reading a page called "nuking the mosquito" where they give very complex proofs for very simple results. The proof I want to talk ...
2
votes
1answer
37 views

Proposition 1A.1 in Hatcher's Algebraic Topology

In the proof of the quoted Proposition, we have a connected graph $X$ and a sequence of subgraphs $X_0 \subset X_1 \subset \cdots$ such that $\cup_i X_i$ is both open and closed. Then Hatcher deduces ...
2
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3answers
50 views

How to determine value from willingness to pay?

I use the British pounds symbol instead of dollars because $ conflicts with Mathjax. Source: p 296, The Legal Analyst, Ward Farnsworth "... one time in a thousand we do lose the film; if you’re ...
0
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1answer
48 views

Proposition I.6.8 in Hartshorne

In the context of the quoted proposition arises the following question. Let $X$ be an abstract nonsingular curve (as defined in p. 42), $P \in X$ and let $\phi: X-P \rightarrow \mathbb{P}^n$ a ...
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1answer
39 views

Proposition I.6.7 in Hartshorne

Proposition (I.6.7,HAG): Every nonsingular quasi-projective curve $Y$ is isomorphic to an abstract nonsingular curve. The first paragraph of the proof establishes a bijective map $\phi: Y \rightarrow ...
0
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1answer
60 views

Hartshorne Theorem I.5.3

This question concerns a reduction argument that occurs in the proof of Theorem I.5.3 in Hartshorne. In particular, let $Y$ be an affine variety of dimension $r$ in $\mathbb{A}^n$. Then by (4.9) $Y$ ...
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1answer
50 views

Weibel “Introduction to homological algebra” Main Theorem 4.4.16

I can't understand the proof of Main Theorem 4.4.16 from Weibel's book "An Introduction to homological algebra". The Theorem states Let $R$ be a local noetherian commutative ring, then $R$ is ...
2
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1answer
41 views

Obscure inequality in a passage of a proof (maximal ergodic theorem)

Look at the following excerpt from the book "Einsiedler and Ward- Ergodic Theory, with a view towards Number Theory": I don't understand why the inequality in the red box is valid. Maybe the ...
1
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1answer
29 views

Question in Fulton and Harris regarding induced representation.

I'm confused by the following paragraph: I don't see why $g\cdot W$ depends only on the left coset $gH$. What does he mean precisely by that? Why is it true that $gh\cdot W = g\cdot(h\cdot W) = ...
1
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1answer
53 views

Groups of order 8 proof

I understand the solution to these questions I was just wanting to confirm that the solution to Q10 excludes the possibility that $y =x^2$ ( and hence the proof is not complete) as it uses the ...
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5answers
80 views

What is the limit of $\log_k(k^a + k^b)$ for $k \to +\infty$?

I'm not very good with analysis (I never studied it) but because of my "work" on other topics of mathematics I came to this problem. $$\lim_{k \to +\infty }\log_k(k^a + k^b)=\max(a,b)$$ I'm sure ...
0
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1answer
32 views

The topology of the restriction of a metric is the restriction of the topology.

I'm reading this proof of the following claim. Let $(X,d)$ be a metric space and $(Y,d')$ a subspace of $X$. If $(X,T)$ is the topology induced by $d$ and $(Y,T')$ the typology induced by $d'$, ...
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1answer
167 views

Hartshorne Theorem 8.17

I can't understand the proof of theorem 8.17 from Hartshorne's "Algebraic Geometry". Namely, he says that we have an exact sequence $$ 0 \to \mathcal J'/\mathcal J'^2 \to \Omega_{X/k} \otimes ...
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1answer
38 views

Explain what the teacher did, convergence of improper integral

I'd like someone to explain what the teacher did, because I'm not sure I understand. Basically, the question is for which values of $p$, does the integral $$\int_{1}^{\infty} \frac{dt}{t \log ...
2
votes
1answer
70 views

Groups product formula's proof

I'm trying to understand the proof of the following theorem: Theorem 2.20 (Product Formula). If $S$ and $T$ are subgroups of a finite group $G$, then $$|ST|\, |S \cap T| = |S|\,|T|.$$ Remark. ...
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2answers
37 views

Uniqueness of Initial Object, or why must a morphism from an object to itself be the identity?

The definition of an initial object in a category $\mathscr{C}$ is defined as an object that only has one map going to each object in $\mathscr{C}$. A basic result supposedly says that any two initial ...
0
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0answers
38 views

Noetherian rings/Hilbert's Basis Theorem

So I'm studying the proof of Hilbert's Basis Theorem - we've shown that $λ(I)$ is an ideal of $R$ and and then it says "Since R is Noetherian, we have $λ(I) = \sum\limits_{i=1}^k s_iR$ for some $s_1, ...
0
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1answer
48 views

Why is this proof valid - inverse function theorem

Question from worksheet, I don't fully understand the solution the teacher gave. Question: let $S$ be the set of symmetric positive definite matrices of dimension $n$x$n$. Let $T: S \to S$, ...
1
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1answer
60 views

Proving that every set in the Borel $\sigma$-algebra on $\mathbb{R}$ is Lebesgue-Stieltjes Measurable

I am reading through Richard Bass's Real Analysis and on page 25 we have the following proposition, using $\alpha$ is a Stieltjes function and $m^*$ is the Lebesgue-Stieltjes outer measure on ...
3
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2answers
52 views

All homomorphisms from a simple ring to a non-zero ring are injective

Let $R$ be a simple ring and $T$ be a non-zero ring. Let $f\colon R \rightarrow T$ be a ring homomorphism. Show $f$ is injective. Proof: $\ker f \lhd R$, so $\ker f=R$ or $\ker f=\{0\}$. If $\ker ...
2
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1answer
51 views

Proof of Pocklingtons theorem: why do we have $\text{ord}_n(a) \mid (q-1)$ but $\text{ord}_n(a) \nmid (n-1)/p_i$?

Proof of Pocklingtons theorem: why do we have $\text{ord}_n(a) \mid (q-1)$ but $\text{ord}_n(a) \nmid (n-1)/p_i$ ? And why does $p_i^{\alpha_i} \mid \text{ord}_n(a)$ ? I see that ...
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3answers
113 views

Proof of k'th power theorem: $x^k \equiv a \pmod n$ has a solution $\iff$ $a^{\phi(n)/\gcd(k, \phi(n))} \equiv 1 \pmod n$.

Proof of k'th power theorem: $x^k \equiv a \pmod n$ has a solution $\iff$ $a^{\phi(n)/\gcd(k, \phi(n))} \equiv 1 \pmod n$, where $n$ has a primitive root. I have proven the following theorem ...
2
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1answer
50 views

Confused About Step in Proof of Divergence of $\sum \frac{1}{p}$

I was going through the number theory text by Ireland and Rosen, and was following the proof of the divergence of the sum of reciprocal primes. But I came across a step unclear to me. The proof so ...
2
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1answer
56 views

Same characteristic polynomial $\iff$ same eigenvalues?

This proves: Similar matrices have the same characteristic polynomial. (Lay P277 Theorem 4) I prefer http://math.stackexchange.com/a/8407/53259, but this proves that they have the same eigenvalues. ...
0
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1answer
80 views

Any $2\times 2$ complex matrix A is similar to one of these three: (See first line of the question)

(i) : $\left(\begin{array}{ll} \lambda_{1} & 0\\ 0 & \lambda_{2} \end{array}\right)$, (ii) : $\left(\begin{array}{ll} \lambda & 0\\ 0 & \lambda \end{array}\right)$, (iii) : ...