azimut
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What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)
101 votes

As a child, the Fibonacci numbers $$1,\; 1,\; 2,\; 3,\; 5,\; 8,\; 13,\; 21,\; 34,\; 55,\;\ldots$$ were very fascinating to me. They are named after the the Italian mathematician Fibonacci, who ...

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Why is the localization at a prime ideal a local ring?
Accepted answer
72 votes

The localization $A_\mathfrak{p}$ is given by all fractions $\frac{a}{b}$ with $a\in R$ and $b\in R\setminus\mathfrak{p}$. So $\mathfrak{p}A_\mathfrak{p}$ consists of all fractions $\frac{a}{b}$ with $...

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A ring with few invertible elements
41 votes

Step 1: The characteristic of $A$ is $2$ (Credit for this observation goes to Jyrki Lahtonen) The mapping $x\mapsto -x$ is an involution on $A^\times$. Since $\lvert A^\times\rvert = 2^n - 1$ is odd,...

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Factorize $(x+1)(x+2)(x+3)(x+6)- 3x^2$
33 votes

I'm assuming that you are looking for a factorization of the polynomial $$ f = x^4 + 12x^3 + 44x^2 + 72x + 36 $$ in $\mathbb Q[x]$. By the Lemma of Gauss and the fact that $f$ is monic, this is the ...

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Can a collection of points be recovered from its multiset of distances?
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31 votes

This is a really nice question! Counterexample for $n=6$ The sets $$\{0,1,4,5,11,13\}\\\{0,1,2,6,10,13\}$$ are affinely inequivalent, but the multiset of differences is in both cases $$1^2\cdot 2\...

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Examples of mathematical results discovered "late"
31 votes

Informally, the historical definition of an Archimedean solid requires that all its faces are regular polygons and that each vertex of the solid locally looks the same. Since the ancient Greeks it was ...

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Classification of groups of order 30
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28 votes

Let $G$ be a group of order 30 and let $n_p$ be the number of its $p$-Sylow subgroups. By Sylow, $n_3\in\{1,10\}$ and $n_5\in\{1,6\}$. It is not possible that $n_3 = 10$ and in the same time $n_5 = 6$,...

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All natural numbers are equal.
28 votes

Hint: The smallest case where your "theorem" is obviously wrong is the set $\{1,2\}$. Now check every step of the "proof" with this example. What goes wrong?

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Examples of mathematical results discovered "late"
26 votes

The theorem that $$\text{PRIMES is in P}$$ was proven only in 2002 (via the AKS primality test). The algorithm certainly isn't rocket science, making it quite surprising that this wasn't found way ...

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$A_4$ has no subgroup of order $6$?
23 votes

By looking at the possible cycle types, we see that $A_4$ consists of the identity element (order $1$), $3$ double transpositions (order $2$) and $8$ $3$-cycles (order $3$). Assume that $A_4$ has a ...

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Show that that $\lim_{n\to\infty}\sqrt[n]{\binom{2n}{n}} = 4$
Accepted answer
21 votes

Hint: By induction, show that for $n\geq 2$ $$\frac{4^n}{n+1} < \binom{2n}{n} < 4^n.$$

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Using $\bigvee$ and $\bigwedge$ instead $\exists$ and $\forall$
20 votes

I try to give you an argument why this notation makes sense. Consider $$\displaystyle\bigvee_x A(x)$$ as an infinite version of $\vee$. For example, if $x$ comes from a countable set $\{x_1,x_2,x_3,\...

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Is a finitely generated projective module a direct summand of a *finitely generated* free module?
19 votes

To completely understand the +1 answer of Martin Brandenburg, I had to add a few details for myself. I decided to document the result in this answer: Let a finite generating system of $P$ be given by ...

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Prove that a set of nine consecutive integers cannot be partitioned into two subsets with same product
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18 votes

Hint Show that none of the 9 consecutive numbers can be a multiple of $13$. Denote the product of the 9 consecutive numbers by $a$. Show that $a$ must be square. Look at the remainder of $a$ modulo $...

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Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$?
18 votes

For such abstract questions, it is important that you stick to the definitions of the involved notions. By definition, $A$ is a subset of $B$ if every element contained in $A$ is also contained in $B$...

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Does the map $f:\emptyset\longrightarrow \{0\}$ exist?
Accepted answer
17 votes

There is exactly one such function, which is the empty set. Explanation: By definition, functions $f : A \to B$ are certain subsets of the cartesian product $A\times B$ (for the defining property, ...

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Show that any abelian transitive subgroup of $S_n$ has order $n$
15 votes

The following solution only needs basic group theory. Let $G$ be an transitive abelian subgroup of $S_n$. By transitivity, for each $i\in\{1,\ldots,n\}$ there is a $\sigma\in G$ such that $\sigma(1) =...

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Inverse of sparse matrix is not generally sparse
15 votes

First of all, as far as I know there is no precise definition of a sparse matrix. The word sparse is used for a series $(A_n)_{n\in\mathbb{N}}$ of $n\times n$ matrices whose fraction of non-zero ...

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What does the semicolon ; mean in a function definition
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14 votes

There is no hard mathematical difference between the comma (,) and the semicolon(;). The semicolon is used sometimes to optically separate some variable group. So the semicolon is not more than a ...

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Find an expression for the number of edges of $L(G)$ in terms of the degrees of the vertices of $G$.
Accepted answer
14 votes

I think the given proof is a bit confusing. Try this one: An edge in $L(G)$ is a $2$-set $\{e_1,e_2\}$ of edges in $G$ with which are adjacent to a common vertex $v$. This vertex $v$ is uniquely ...

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How to convert $\pi$ to base 16?
13 votes

For the particular base of $16$, there is this remarkable formula: $$\pi=\sum_{n=0}^\infty \left(\frac{4}{8n+1}-\frac{2}{8n+4}-\frac{1}{8n+5}-\frac{1}{8n+6}\right)\frac{1}{16^n}$$ It allows the ...

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Examples of abelian subgroups of non-abelian groups.
13 votes

The smallest non-abelian group is the symmetric group $S_3$ of order $6$. So all its proper subgroups are abelian (the trivial subgroup, three subgroups of order $2$ and one subgroup of order $3$).

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Show that $x^{n-1}+\cdots +x+1$ is irreducible over $\mathbb Z$ if and only if $n$ is a prime.
12 votes

The key is the repeated application of the geometric summation formula $$ \sum_{i=0}^{z-1} q^i = \frac{q^z - 1}{q-1}. $$ Plugging in $z=n, q=x$ and assuming $n = kl$, we get $$p(x) = x^{n-1} + x^{n-2}...

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Determine the degree of the splitting field for $f(x)=x^{15}-1$.
Accepted answer
12 votes

a) Algebraic extensions of $\mathbb{R}$ have degree $1$ or $2$. Since not all zeros of $x^{15} - 1$ are in $\mathbb{R}$, we get $[L : \mathbb{R}] = 2$. b) Let $\zeta$ be a primitive $15$th root of ...

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Is $\mathbb{Z}^2$ cyclic?
12 votes

No, it isn't. For all $a,b\in\mathbb{Z}$ it holds that $\langle (a,b)\rangle = \{(ka,kb) \mid k\in\mathbb{Z}\} \neq \mathbb{Z}^2$. So $\mathbb{Z}^2$ is not generated by a single generator and hence ...

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Example of group homomorphism $f: G \to G$ that is injective, but not surjective.
12 votes

$f : (\mathbb{Z},+) \to (\mathbb{Z}, +), \quad x\mapsto 2x$

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At most one subgroup of every order dividing $\lvert G\rvert$ implies $G$ cyclic
11 votes

Let $a_d$ denote the number of elements of order $d$ and $u_d$ the number of cyclic subgroups of order $d$. Then $a_d = \varphi(d) u_d$, where $\varphi$ is the Euler phi function. Reason: A cyclic ...

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Probability of rolling at least one 6 while rerolling 1's
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11 votes

The best strategy is to reroll any rolled 1, of course. So using that strategy, what is the chance that you do not end up at a 6 with a single dice? The probability for an instant 6 is $1/6$, and for ...

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Do groups, rings and fields have practical applications in CS? If so, what are some?
11 votes

Two important applications which are involved at every phone call on your mobile phone: Coding theory (adding redundancy to the information, such that occuring errors can be compensated) and ...

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Is any subgroup of a direct product isomorphic to a direct product of subgroups?
Accepted answer
11 votes

This is based on the comment of @Derek Holt: A counterexample is given by the subgroup $$V = \{(g,h) \in S_3\times S_3 \mid \operatorname{sgn}(gh) = +1\}$$ of $S_3\times S_3$. Reason: Since $V$ is ...

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