If you already have a proof for some result, but want to ask for a different proof (using different methods).

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0
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1answer
325 views

Norm of the product of two regular ideals of an order of an algebraic number field

Let $K$ be an algebraic number field of degree $n$. Let $\mathcal{O}_K$ be the ring of algebraic integers. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank ...
2
votes
1answer
110 views

Factor ring by a regular ideal of a one-dimensional Noetherian domain

Let $A$ be a one-dimensional Noetherian domain. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is a finitely generated $A$-module. It is well-known that ...
4
votes
2answers
120 views

On the existentence of an element of a group whose order is the LCM of orders of given two elements which are commutaive

I came up with the following proposition. Proposition Let $G$ be a group. Let $x, y$ be elements of finite order in $G$ such that $xy = yx$. Let $n$ be the order of $x$. Let $m$ be the order of $y$. ...
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2answers
94 views

Determination of the prime ideals lying over $2$ in a quadratic order

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, $D$ its discriminant. By this question, $1, \omega = \frac{(D + \sqrt D)}{2}$ is a basis of $R$ as a $\mathbb{Z}$-module. Let ...
-4
votes
1answer
115 views

Determination of the prime ideals lying over an odd prime in a quadratic order

We need some notation before we state the problem. Let $K$ be a quadratic number field, $d$ its discriminant. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module of ...
-1
votes
1answer
142 views

Conductor of a quadratic order

We need some definitions to state the problem. Let $B$ be a commutative ring, $A$ its subring. We denote by $(A : B)$ the set $\{x \in B | xB \subset A\}$. $(A : B)$ is an ideal of $B$. It is ...
-1
votes
1answer
81 views

Condition for $a, b + \omega$ to be the canonical basis of an ideal of a quadratic order

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, $D$ its discriminant. By this question, $1, \omega = \frac{(D + \sqrt D)}{2}$ is a basis of $R$ as a $\mathbb{Z}$-module. Let $I$ be a ...
2
votes
3answers
47 views

Commutativity or Distributivity - Which One to Use to DEFINE Multiplication of Negative Numbers?

It's easy to calculate $3 \times (-4)$, using the meaning of multiplication: $3 \times (-4)=(-4)+(-4)+(-4)=-12$. But it's not the case about $(-4)\times 3$! To DEFINE $(-4)\times 3$ we can choose ...
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votes
1answer
130 views

Canonical basis of an ideal of a quadratic order

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, $D$ its discriminant. I am interested in the ideal theory on $R$ because it is closely related to the theory of binary quadratic forms ...
2
votes
1answer
68 views

The sequence $H_n-\ln(n)$ converges

Is there a proof that the sequence $\displaystyle \sum_{k=1}^n \frac{1}{k}-\ln(n)$ converges that doesn't use integrals?
1
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1answer
72 views

Roots of the characteristic polynomial of a symmetric matrix

I'm looking for a proof (using basic tools : definition of the characteristic polynomial and its basic properties) of the following fact : The roots of the characteristic polynomial of a symmetric ...
2
votes
1answer
109 views

Necessary and sufficient conditions for a polynomial $f(x) \in \mathbb{Z}[x]$ to be irreducible.

This is not for homework, and I am not totally convinced I understand the question entirely. Also, I'm not allowed to use the fact that $\mathbb{Z}[x]$ is a UFD. The question asks Show that ...
3
votes
0answers
65 views

Three old chestnuts in elementary geometry: is there a unified perspective? [duplicate]

Coxeter and Greitzer, in their excellent Geometry Revisited, list a few "hardy perennials" in elementary geometry: tough problems solvable by elementary methods. Their problem number 4 (on page 26 in ...
1
vote
3answers
63 views

A problem on finding dy/dx

If $a+b+c=0$ and $$y=\frac{1}{x^b+x^{-c}+1}+\frac{1}{x^c+x^{-a}+1}+\frac{1}{x^a+x^{-b}+1}$$then $\frac{dy}{dx}$=? The only way which I can think of solving this is by differentiating each term. ...
-1
votes
2answers
452 views

Conditions for a real binary quadratic form to be positive definite

Since this question was heavily downvoted, I would like to change the presentation of the question as follows. I hope those of you who downvoted this question would be satisfied with the change. In ...
0
votes
1answer
119 views

On a certain basis of an order of a quadratic number field

Let $K$ be an algebraic number field of degree $n$. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank $n$. Let $\alpha_1, \cdots, \alpha_n$ be a basis of $R$ ...
0
votes
1answer
42 views

Asking for an analytical proof

Calculate the following integral for $n\in\mathbb {N}$" $$\int_{1}^{n+1} \frac {\{x\}^{[x]}}{[x]} dx$$ :lol: Where $[x]$ is the largest integer $\leq x$ and $\{x\}=x-[x]$ Well my attempt was using ...
3
votes
5answers
96 views

Induction: $n + 3 < n!$ for all $n>3$

I have a proof that I am trying to prove and I am getting stuck at the inductive hypothesis. This is my theorem: For all real numbers $n>3$, the following is true: $n + 3 < n!$. I have ...
0
votes
1answer
69 views

Starting Bisection Proof of Extreme Value Theorem

I am having difficulties beginning a proof for the following statement: Use a proof strategy of bisection to prove that every function $f:[a,b] \to \mathbb{R}$ that is not bounded above is ...
2
votes
0answers
145 views

Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
4
votes
2answers
169 views

If $n$ is squarefree, $k\ge 2$, then $\exists f\in\Bbb Z[x_1,\dots,x_k] : f(\overline x)\equiv 0\pmod n\iff \overline x\equiv \overline 0\pmod n$

Starting from this question, we set $n=k=2$ and use the function $f\in\Bbb Z[x,y]$ where $f(x,y)=x\cdot y+x+y$, then the proofs applied to that question satisfy this case. Note that for $k=1$ the ...
0
votes
0answers
94 views

Proving that morphism of sheaves is iso iff induced morphism on stalks is iso

Is the following proof sound/does anyone have another more elegant (categorical) proof? The direction $\Rightarrow$ is obvious the "family of stalks"-functor is a functor and functors take isos to ...
20
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9answers
5k views

What is the simplest proof of the pythagorean theorem you know? [duplicate]

Maybe enough so to explain it to children.
0
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1answer
38 views

Finding a special similar matrix

I'm looking for a simpler proof for the following exercise : I have $5$ matrices, denoted by $M_i$, of size $2\times 2$ (with real coefficients) and I need to find, for each matrix $M_i$, an ...
9
votes
7answers
929 views

If $xy$ and $x+y$ are both even integers (with $x,y$ integers), then $x$ and $y$ are both even integers

The title statement can be proven using the contrapositive, note that $x$ odd or $y$ odd means that at least one of $x\cdot y,x+y$ is odd. Is there a way to prove the statement directly? To ...
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2answers
149 views

Alternative proofs of the remainder theorem for polynomials

The theorem I've been tasked with proving is that for any polynomial function $f:\mathbb{R} \to \mathbb{R}$ and any $a \in \mathbb{R}$ there exists some polynomial $g:\mathbb{R} \to \mathbb{R}$ and ...
7
votes
5answers
226 views

Different ways to prove $\sqrt p$ irrational for $p$ prime.

I know this fact can be proved by contradiction(reductio ad absurdum) but please give proofs by different methods.
2
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2answers
36 views

Standard inductive problem

Question: Prove that $2^n \geq (n+1)^2$ for all $n \geq 6$. I have tried to prove this below and I'm interested if my method was correct and if there is a simpler answer since my answer seems ...
3
votes
1answer
132 views

Proofs regarding Continuous functions 2

I need verification for this proof: Q: Suppose $f: (0,1)\rightarrow \mathbb{R}$ is defined by $f(x) = \begin{cases}\frac{1}{n} & \text{if }\text{x is rational with x} = \frac{m}{n}\text{ in ...
1
vote
1answer
81 views

Proofs regarding Continuous functions 1

Q: Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a bounded function (that is, there exists some $M\geq 0$ so that $|f(x)|\leq M$ for all $x\in\mathbb{R}$). Define a new function ...
0
votes
1answer
109 views

Alternatinve proof for the principle of the Iterated Suprema

The back of the book gave a proof similar to the proof here Proving principle of the Iterated Suprema, but I proved it following way before I checked the back of the book. Could some one verify this ...
0
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1answer
68 views

Examples of sets which measure cannot be obtained by discretisation

I started reading "An introduction to measure theory" by Terence Tao. On page 23 on a pdf reader (pg 7 in the actual document), we are asked to think of an example of a set $E\subset$ ...
3
votes
1answer
89 views

On Elements of $p$th Row in n Pascal's Triangle (For Prime $p$)

If $p$ is a prime number, in Pascal's triangle all the terms in the $p$th row - except the 1s - are multiples of $p$ . It's easy to prove this property using the formula for $\binom{p}{k}$. Is there ...
1
vote
0answers
39 views

Ineqality regarding LCM of $1, 2, \ldots, n$

While going through F. Beukers proof of irrationality of $\zeta(3)$ I found the inequality $d_{n} < 3^{n}$ for all sufficiently large values of $n$ where $d_{n}$ denotes the LCM of all the numbers ...
1
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0answers
69 views

Maximally Consistent Set (Proof by Contradiction)

Yesterday, I asked about feedback for a proof of the following theorem For all $\phi$, $\phi \in \Gamma^{*}$ if and only if $\Gamma^{*} \vdash \phi$. My main concern was the first part $(\to)$, which ...
0
votes
0answers
43 views

On Maschke's theorem: proof with matrix skills

In page 4 by Isaacs' book "character theory of finite groups", there is a proof for Maschke's theorem. In page 48 by Curtis and Reiner's book Representation theory of finite groups and associative ...
1
vote
4answers
150 views

Elegant or elementary evaluation of $\lim\limits_{x\to 0} \left( \frac{1}{x}-\frac{1}{\sin(x)} \right) $ [duplicate]

I give math tutoring and was wondering about the following limit. I found the answer but I was wondering if someone has a nicer explanation than the one I am giving where I use L'Hôpital's rule twice. ...
0
votes
2answers
68 views

prove there exists $x$ in ${\mathbb R}$ using the completeness axioms

Let $a, b \in {\mathbb R}$ with $a < b$. Prove that there exists $x\in {\mathbb R}$ which is NOT a rational number such that $a < x < b$. This is what i have at the moment.. It does not have ...
4
votes
3answers
158 views

Alternative Creative Proofs that $A_4$ has no subgroups of order 6

Since I've been so immersed in group theory this semester, I have decided to focus on a certain curious fact: $A_4$ has no subgroups of order $6$. While I know how to prove this statement, I am ...
0
votes
1answer
109 views

Showing that $a_n \not \to 17$ implies a subsequence $a_{n_k}$ that is $\epsilon$ far from $17$ for some $\epsilon > 0$

I want to check my proof for this question: Suppose a sequence {$a_n$} does not converge to 17. Prove that there exists some $\epsilon$ > 0 and a subsequence {$a_{nk}$} so that $|a_{nk} - 17|$ > ...
3
votes
1answer
138 views

Help with a lemma of the nth root (without the binomial formula)

I have no idea of how to solve it. I would appreciate if someone gives me a hint, please. Definitions Let $\,x^{1/n}:= sup\{\, y \in \mathbb{R}: y\ge0 \text{ and } y^n\le x\, \}$ Lemma: Let ...
2
votes
2answers
204 views

Verification of proof of the Sequence of Arithmetic Theorem

Suppose $\left\{b_{n}\right\}$ is a sequence of real numbers which converges to $M$, so that $b_{n} \neq 0$ for each $n$, and $M \neq 0$. Prove that the sequence $\{ \frac{1}{b_n} \}$ converges to ...
0
votes
2answers
113 views

$n!\sum_{k=1}^n \frac{a_{k}}{k!}$ is always integer.

$\displaystyle n!\sum_{k=1}^n \frac{a_{k}}{k!}\in \mathbb{Z}$ where $n,a_k$ and $k$ are integers. I know the proof by induction. Is there any other technique to prove it? Thank you.
2
votes
1answer
149 views

Is there a Schur-Zassenhaus-free proof that $\Phi(G)$ cannot contain a Sylow subgroup of $G$?

As we know, the Frattini subgroup of a finite group G can not contain a Sylow subgroup of G, but if we want to prove this, we need the Schur-Zassenhaus theorem. What I want to know is if there is a ...
5
votes
1answer
165 views

Alternative argument - set theory problem

I tried to solve the following problem: Let $ \mathcal{F}$ be a nonempty family of sets with the following properties: (a) If $ X \in \mathcal{F}$, then there are some $ Y \in \mathcal{F}$ ...
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2answers
2k views

Prove that $\sup(A+B) = \sup(A) + \sup(B)$ and why does $\sup(A+B)$ exist?

We want to show that $\sup(A)+\sup(B)$ is the least upper bound of the set $A + B$. First, we need to show that $\sup(A) + \sup(B)$ is an upper bound for the set $A + B$. Indeed, if $z\in A + B$, then ...
4
votes
4answers
185 views

Prove $3^n \ge n^3$ by induction

Yep, prove $3^n \ge n^3$, $n \in \mathbb{N}$. I can do this myself, but can't figure out any kind of "beautiful" way to do it. The way I do it is: Assume $3^n \ge n^3$ Now, $(n+1)^3 = n^3 + ...
2
votes
1answer
247 views

Proof for $-\sup(A) = \inf(-A)$

Let $A$ and $-A = \{ -x \mid x \in A \}$ be two bounded sets. I have to prove that $-\sup(A) = \inf(-A)$, i did it in the following way and wish to know if it is sufficient: $ \exists x\in A$ such ...
5
votes
1answer
207 views

Dividing Squares Fails to Invoke Contradiction: Two Elementary Divisibility Proofs

$x^2 \text{ is even } \iff x \text{ is even } \tag{Thm 3.12, P76}$ $\text{ Let } x, y \in \mathbb{Z}. \text{ Then } x \;\& \; y \text{ are of the same parity } \iff x + y \text{ is even.} \tag{Thm ...
2
votes
1answer
118 views

Every finite group is finitely generated. - alternate proof

Is that's all? Thank you. :-) A group $H$ is called finitely generated if there is a finite set $A$ such that $H = \left \langle A \right \rangle$ . Prove that every finite group is finitely ...