1
vote
2answers
61 views

Conjugation (Group vs. Algebraic)

I am just starting to learn about groups, and the concept of conjugation came up. I was wondering what the relationship was, if any, between conjugation in the group sense and conjugation in the ...
0
votes
1answer
27 views

Every non-empty subset of the integers which is bounded above has a largest element.

I was reading a proof about every non-empty subset of the integers which is bounded above has a largest element, but i have troubles in one step. Here is the proof: Since $S$ is a non-empty subset of ...
1
vote
1answer
23 views

Proof, that equation decribes trace of curve, which is supposed to be simple

The equation, representing the trace of the curve $$ \varphi(x) = (\cos^3(t), \sin^3(t)) $$ is $1 = x^{\frac{2}{3}} + y^{\frac{2}{3}}$. Proof: Let $(x,y) = (\cos^3 t, \sin^3 t)$, then $x^{1/3} = ...
0
votes
1answer
14 views

Prove $x_1$ is at least a $k$-fold root of polynomial $p$ if and only if $p(x_1) = p^{'}(x_1) = \dots p^{(k-1)}(x_1) = 0$?

Suppose $p: \mathbb R \rightarrow \mathbb R$ is a polynomial given by $x \mapsto a_nx^n + \dots a_1 x_1 + a_0$. How do I prove $x_1$ is at least a $k$-fold root of $p$ if and only if $p(x_1) = ...
0
votes
1answer
18 views

Tangent line at $x_1$ to polynomial curve $p(x)$ of degree at least $2$ implies $x_1$ is a double root of $p(x) - p^{'}(x_1)(x-x_1)$?

Tangent line at $x_1$ to polynomial curve $p(x)$ of degree at least $2$ implies $x_1$ is a double root of $p(x) - p^{'}(x_1)(x-x_1)$ ?. Suppose I have a polynomial function $p(x): \mathbb R ...
0
votes
2answers
39 views

Expansion of Cyclotomic polynomial on input $x+1$

I've been wondering why $$x^{p-1}+x^{p-2} + \dots + x + 1$$ expands to $$x^{p-1} + \binom p 1x^{p-2} + \binom p 2x^{p-3} + \dots + \binom p 1$$ when substituting $x$ for $x+1$ ? Can someone clarify ...
11
votes
0answers
166 views

Denesting radicals like $\sqrt[3]{\sqrt[3]{2} - 1}$

The following result discussed by Ramanujan is very famous: $$\sqrt[3]{\sqrt[3]{2} - 1} = \sqrt[3]{\frac{1}{9}} - \sqrt[3]{\frac{2}{9}} + \sqrt[3]{\frac{4}{9}}\tag {1}$$ and can be easily proved by ...
4
votes
0answers
72 views

Irreducibility of some polynomial

Let $p(x) = (1+ \cdots +x^k)^2 + (1+ \cdots +x^k) + 1$, for some $k \geq 2$ fixed. I would like to know if $p(x)$ is irreducible in $\mathbb{Q}[x]$.
2
votes
0answers
40 views

How “separable” (not in that sense) is a polynomial?

Since "separable" is used for different meaning in separable polynomial and separation of variable, I am having trouble searching for anything related to my question. So I hope someone can help with ...
19
votes
2answers
2k views

Why are higher-degree polynomial equations more difficult to solve?

I am confused about the significance of the powers on equations. For example, in $ax = b$, intuitively $b$ is a value $x$ multiplied $a$ times. In $ax + b = c$, $c$ is a value $x$ multiplied $a$ times ...
0
votes
1answer
43 views

Significance of higher order equations [closed]

I am confused about the significance of the powers on equations. For example, in ax = b, intuitively b is a value x multiplied a times. In ax + b = c, c is a value x multiplied a times added to by b. ...
2
votes
2answers
427 views

How can I find the roots of a quartic equation, knowing one of its roots?

I need to decompose (in $\Bbb{C}[x]$) the function $$ f(x) = x^4 + 4x^3 - 4x^2 + 24x + 15 $$ in its simplest form, knowing that $1 - 2i$ is one of its roots. Any ideas?
0
votes
0answers
32 views

Subset Sum represented as a perfect number

Can we form a set of $29$ distinct integer elements such that every subset of elements possible has a sum which is a perfect power? A perfect power is a positive integer that can be represented a p^q ...
0
votes
3answers
61 views

Factorizing degree four polynomial

Please help me to factorize $x^4+3x^3+6x+10$ completely over $\Bbb{Q}$ I have tried setting the equation equal to $(x+a)(x^3+bx^2+cx+d)$ and comparing the coefficients, but I seem to have too many ...
0
votes
0answers
39 views

The elegant expression in terms of gcd and lcm - algebra - (2)

Definition: suppose a quantity $P$ is identified by $$ \frac{P}{k}\simeq \frac{P}{k}+1 $$ what we mean is that $$ P= 0\pmod{k}. $$ That means that when $P \to P+k$, then $$ \frac{P}{k}\to ...
2
votes
1answer
43 views

The elegant expression in terms of gcd and lcm - algebra

Given three positive integer numbers $k_1$, $k_2$, $k_3$, we may denote their greatest common divisor(gcd) by $\gcd(k_i,k_j)\equiv k_{ij}$ for gcd of a two pair of number $k_i,k_j$. ...
1
vote
1answer
29 views

How many n-tuple to genereate zero from some random several variable equation that use constant power and a variable as the base?

Define algebraic number tuple as the aphabetical order sequence of variables that use in the equation. How many algebraic number n-tuple (x,y,z,...) are able to genereate zero to input into a several ...
0
votes
1answer
28 views

how to solve the gaussian brackets?

could someone please explain how the following expression has been reshaped,particularly the gaussion brackets. $(-1)^n * n +(-1)^{n-1} * \lceil \frac{n-1}{2} \rceil$ has been reshaped to $(-1)^n ...
1
vote
0answers
175 views

Solving an 8th degree polynomial

I know that through the Abel Ruffini Theorem the general solution to a polynomial of degree five or more cannot be found explicitly. But are there are any other ways to find the roots of such a ...
1
vote
2answers
160 views

Algorithms for factoring multivariate polynomials

I am wondering if there are any algorithms to factor polynomials in multiple variables, when you know that the factors are other polynomials with rational or integer coefficients. I know you have the ...
1
vote
0answers
67 views

Problem in solving quartic equations

There is only one step in Descartes' method for solving the quartic that confuses me. Consider the depressed quartic $x^4+px^2+qx+r=0$, note I am assuming that the coefficients $p,q,r\in\mathbb{C}$. ...
0
votes
1answer
68 views

What will be the time complexity of insertion if a queue is implemented using two stacks?

A Queue could be implemented using two Stacks. So what will be the time complexity for insertion and deletion in this queue? Thanks in advance.
2
votes
2answers
170 views

Does Fermat's Little Theorem work on polynomials?

Let $p$ be a prime number. Then if $ f(x) = (1+x)^p$ and $g(x) = (1+x)$, then is $f \equiv g \mod p$? I'm trying to prove that for integers $a > b > 0$ and a prime integer $p$, ${pa\choose b} ...
1
vote
2answers
78 views

Polynomials through successive differences

Let $h_0:\Bbb{N}\rightarrow\Bbb{N}$ be any function. Define recursively, for $m>0$, $$h_{r+1}(m)=h_r(m)-h_r(m-1).$$ Suppose that for some $k>0$ we have $h_k(m)\equiv d$ constant. Is this ...
1
vote
3answers
112 views

How can efficiently derive $x$ and $y$ from $z$ where $z=2^x+3^y$.

How can efficiently derive $x$ and $y$ from $z$ where $z=2^x+3^y$. Note. $x$,$y$ and $z$ are integer values and $z$ is $4096$ bits integer or even more. For all $z>1$. And if equation be ...
0
votes
1answer
72 views

Finding Percentage Contribution of a Variable in an Equation

I have an equation, for example: $$ y=a-b+c $$ I am actually confused how exactly to find the contribution of the variables individually to the entire equation. Due to the negative sign, following ...
2
votes
2answers
110 views

About linear transformations

Let $V$ be the real vector space of 2x2 matrices and $End (V)$ the space of all linear transformations of V in V. $$T: V \rightarrow End (V)$$ $$T(A)(B)=AB-BA$$ I have to prove that this is a linear ...
2
votes
4answers
224 views

Why do we have to do the same things to both sides of an equation?

Forgive me in advanced if this is a trivial question. This convention makes perfect sense to me intuitively, but is there any rigorous underpinning to it? I'm beginning to read through an abstract ...
1
vote
4answers
80 views

Proof: Two polynomials $P(x)$ and $Q(x)$ attain same value for every $x \in \mathbb R$ if and only if coefficients $p_i = q_i$ are equal for every $i$

Proof: Two polynomials $P(x)$ and $Q(x)$ attain same value for every $x \in \mathbb R$ if and only if coefficients $p_i = q_i$ are equal for every $i$ I've been thinking how to prove this. I know we ...
-1
votes
3answers
139 views

Group theoretic confusion on different group structures induced on the real line by ordinary multiplication and division

This is a followup to one of my responses to a question by my old teacher Ravi Kulkarni found at A question about groups: may I substitute a binary operation with a function?. I attempted to make a ...
2
votes
1answer
96 views

Associativity of a composition $x ∘ y = xy+\sqrt{(x^2-1)(y^2-1)}$

For many hours I had been stucked at this problem. For the following composition $x ∘ y = xy+\sqrt{(x^2-1)(y^2-1)}$ I have to demonstrate that this composition is associative($(x ∘ y)∘z=x∘(y∘z)$ and ...
12
votes
1answer
345 views

When $\sin x, \cos x$ are $\mathbb{Q}$-linear combinations of square roots

Suppose $x\in\Bbb R$ is such that $$\sin x=\sum_{i=1}^m x_i\sqrt{r_i},\quad \cos x=\sum_{j=1}^n y_j\sqrt{s_j}$$ for some $x_i, r_i, y_j, s_j \in\Bbb Q \ , \ |x_i|=|y_j|=1$. Show that ...
1
vote
1answer
90 views

How expand the domain of the $\log$ function from $\Bbb Z$ to $\Bbb Q$?

Let function $f(N)=\log N$ while $N \in \Bbb N$ One can expand the domain from $\Bbb N$ to $\Bbb Z$ by a clear mathematical approach $$f(N) = \left\{ \begin{array}{l l} \log N & \quad ...
7
votes
4answers
292 views

How to justify the existence of a function, in general?

Maybe I'm too naive in asking this question, but I think it's important and I'd like to know your answer. So, for example I always see that people just write something like "let $f:R\times ...
32
votes
4answers
2k views

Does there exist rational $a,b,c$, such that $\sqrt[3]{1}+\sqrt[3]{2}+\sqrt[3]{4}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$

Let $w = \sqrt[3]{1}+\sqrt[3]{2}+\sqrt[3]{4}$. How to prove that there are no triples $(a,b,c)$, such that $a,b,c \in \mathbb{Q}$; $a \leqslant b \leqslant c$; $(a,b,c)\ne (1,2,4)$; $w = ...
3
votes
0answers
249 views

Recognizing subadditivity

Let $f: (0,\infty) \to \mathbb{R}$ be some continuous function. We say that $f$ is subadditive if the bound \begin{align} f(x+y) \leq f(x) + f(y) \tag{1} \end{align} holds. I was attempting to ...
1
vote
1answer
29 views

Algebra / Equation with 2 or more vaiables

A train moves past a telegraph post and a bridge 264 m long in 8 seconds and 20 seconds respectively. What is the speed of the train? I'm really confused if there are 2 or more variables. Can ...
2
votes
2answers
69 views

Why does $\sum_{k=n_0}^n{P(k)}$ is a polynomial in $n$ of degree $d$?

Let $P(X)$ be a polynomial over $\mathbb{Z}$ of degree $d-1$ and $n_0$ be some constant positive integer. Then why does $\sum_{k=n_0}^n{P(k)}$ is a polynomial in $n$ of degree $d$?
1
vote
1answer
66 views

Localization of a ring which is not a domain

Let $A$ be a ring (commutative with $1$), let $S$ be a multiplicatively closed subset of $A$, i.e $S$ is contained in $A$ , $1\in S$ and $a,b\in S$ implies $ab\in S$, for every $a,b\in A$. Consider ...
4
votes
3answers
276 views

There isn't a product operation that is commmutative on $ \mathbb{R}^{n} $ that satisfies all the field axioms for $ n \geq 3 $.

This proof is broken down into simple easy algebra and vector questions. I would like to discuss different answers and approaches. Please see pg 162-163 on books.google.ca/books?isbn=0387290524 ...
0
votes
1answer
56 views

Simplifying equation from an aggregate one

I have got the 4th order coefficient of $\epsilon$ from the equation (9) from the paper. I got : $$\phi_4+ \ddot \phi_4+\omega_2 \ddot \phi_2 - \Delta \phi_2+g_2 \phi_2^2+2 \phi_1 \phi_3+ 3g_3 ...
0
votes
1answer
99 views

I want to find a closed form of the sum $\sum_{i=1}^n x^{i^2}$

Is there any closed form of the following sum? $\sum_{i=1}^n x^{i^2}$, where $x$ is a variable. Calculating the sum for a first few $n$ does not give any pattern.
3
votes
2answers
1k views

Division algorithm for multivariate polynomials?

We know that if $F$ is a field and $f(X)$ a non-zero polynomial in $F[X]$, then for every polynomial $g(X)$ we can find $q(X),r(X)$ such that $$g(X)=f(X)\cdot q(X)+r(X)$$ with $r(X)$ the zero ...
1
vote
1answer
54 views

Extending a Bijective Map

I have a problem I’m working on that I know the answer to, but I still need some help understanding some things. The problem is as follows: Show that $S = \{ x \in \mathbb{R} \space | \space 0 ...
0
votes
1answer
129 views

Weak direct product

I am just reading the book "Algebra" by Hungerford and on one page it says that if $G_i$ is a family of groups $\forall i\in I$ then $\prod_{i\in I}^{w}G_i\unlhd\prod_{i\in I}G_i$ where ...
4
votes
2answers
1k views

Direct sums and direct products

This question has been in my head for a while. And today it appears again when I am reading Arveson's book on $C^*$-algebras. He says Countable direct products of Polish spaces are Polish. ...
1
vote
1answer
88 views

Simplify the expression

The below expression has three summations (sigmas) and $L$ is a real-matrix and symmetric, $X$ is a real matrix with $n$ rows and $X_{p\mathbb{.}},X_{q\mathbb{.}}$ denote the $p$ and $q$ rows of ...
0
votes
0answers
35 views

Characterization of n-grade, m-variables polynomials $P$ over $\mathbb{R}$, with $P(x_1,\ldots,x_m)\in [0,1]$ if $\forall x_i \in [0,1]$ .

I'll write an introduction. This problem came to me while I was doing an experiment with the prisoners dilemma. I'm codifing the agents behavior in genes (I want to measure the correlation between the ...
0
votes
1answer
189 views

Ratio between two sets of numbers?

I have two sets of numbers. The first set is 1 to 4. The second set is 0 to 190. What is the calculation to get the proportionate number in the second set if a number in the first set is, for example, ...
3
votes
4answers
448 views

Showing that $a+b\sqrt{2}$ is associative over multiplication

I need to show that $a+b\sqrt{2}$ is associative over multiplication. This is what I have so far. I may be taking a wrong route so just please let me know. $$(a+b\sqrt{2})*((c+d \sqrt{2})*(e+f ...