# Questions tagged [roots-of-unity]

numbers $z$ such that $z^n=1$ for some natural number $n$; here usually $z$ is in $\mathbb C$ or some other field

662 questions
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### Conjecture about some group semiring representations ( and roots of unity ).

Let $\Bbb R_+=[0,\infty)$ be a semiring. $\Bbb R_+[C_n]$ is the group semiring formed by the semiring $\Bbb R_+$ and the cylic group $C_n$. Let $\Bbb R_+[X_n]$ be the polynomial semiring. (...
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### Conjecture about some rings and roots of unity. [duplicate]

Let $\Bbb R_{\geqslant 0}[X_n]$ be a polynomial semiring. More precisely $\Bbb R_{\geqslant 0}[X_n]$ are the polynomials of $X_n$ with positive real coefficients with $(X_n)^n = 1$. Let $F(n)$ be ...
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### Roots of unity in CM-field

Let $K$ be a CM-field, ie. a totally imaginary quadratic extension of a totally real number field $F$ and let $p > 2$ be a rational prime. My question simply is Are the $p$-th roots of unity, ...
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### Simple-looking bound on root of unity

I am trying to prove some bound and stuck with the following: If $|n|\leq 3N/4$, then $\left|e^{2\pi in/N}-1\right|\geq\dfrac{n}{N}$ ($n,N$ are integers) How can I prove it?
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### Adjoining two primitive n-th roots

Let $\omega_n$ denote a primitive $n^{th}$ root of unity. If $m$ and $n$ are positive integers with $lcm(m,n)=k$, show that $\mathbb{Q}(\omega_n,\omega_m)=\mathbb{Q}(\omega_k)$. To start, I am aware ...
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### Why does Wolfram Alpha say that $\sqrt{1}=-1$? [duplicate]

Why does Wolfram Alpha say that $\sqrt{1}=-1$? Is this a mistake or what? Can anyone help? Thanks in advance.
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### Is $\sqrt 7$ the sum of roots of unity?

Let $a_n$ and $b_n$ be 2 sequences of $n$ rationals. Is it possible that $\sqrt 7 = \sum_{m=1}^{n} a_m (-1)^{b_m}$ ? Is it possible that $\sqrt{17}$ = $\sum_{m=1}^{n} a_m (-1)^{b_m}$ ? How to ...
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### prove that polynomial has root of unity

Prove that $f=x^n\pm x^m\pm1$ is either irreducible over rationals or has a root which is a of unity. I tried to see what if $x=|r|e^{i\phi}$ but I have no proper result.
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### Evaluating a polynomial at a root of unity?

Let $R = \mathbb{Z}[x]/(x^n+1)$ be the $2n$th cyclotomic ring (for $n$ a power of $2$ in which case $\Phi_{2n}(x) = x^n+1$). Let $g$ be an $n$-dimensional vector chosen at random from $\mathbb{Z}^n$ ...
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### Why is the intersection of Q($\sqrt[n]{a}$) and Q(nth root of unity) Galois? (a>0, n an integer)

With this, I can show the intersection is either Q or Q($\sqrt{a}$). All I have is that their intersection must be real and all subfields of Q($\sqrt[n]{a}$) are of the form Q($\sqrt[d]{a}$) where d|...
For a Galois Theory class I've been asked to find a radical extension with a non-radical subextension (all over $\mathbb{Q}$). So, I'm looking at the splitting field of $x^7 - 1$, namely $\mathbb{Q}(\... 1answer 1k views ### Why can't we just say 1 instead of “unity”? I know this is a soft question of sorts but I am curious why we can't just say "1" instead of "unity," e.g. a root of unity. 2answers 986 views ### Radical extension Let$K=\mathbb{Q}(\sqrt[n]a)$where$a\in\mathbb{Q}$,$a>0$and suppose$[K:\mathbb{Q}]=n$. Let$E$be any subfield of$K$and let$[E:\mathbb{Q}]=d.$Prove that$E=\mathbb{Q}(\sqrt[d]a)$. It's ... 1answer 573 views ### A Trigonometric Sum Related to Gauss Sums This is a problem given to me by fractals on Art of Problem Solving. I couldn't solve it so I'm posting it here for some thoughts on it. Let $$S = \sum_{j = 0}^{\lfloor n/2 \rfloor} \sin\dfrac{2\pi\... 2answers 89 views ### The sum of the first and last k-th roots of unity Let \zeta_k be the first complex primitive k-th root of unity, i.e. \zeta_k = e^{2\pi i/k}. Then the last complex primitive k-th root of unity is given by \zeta_k^{k-1} = \zeta_k^{-1} = e^{-2\... 3answers 169 views ### For which values of k, 0\leq k \leq n-1, is e^{i2πk/n} a primitive nth root of unity? I know that the n-th root of unity is a primitive nth root of unity if, and only if, k is relatively prime to n, but how do you prove it? 1answer 133 views ### Alternating Series Using Other Roots of Unity \sum (-1)^n b_n is representative of an alternating series. We look at whether \sum b_n converges and if b_{n+1}<b_n \forall n\in \mathbb{Z}. What if our alternating series is of the form ... 1answer 359 views ### Roots of unity in a field generated by a root of a polynomial The polynomial x^3 − 2 has no roots in F7 and is therefore irreducible in F7[x]. Adjoin a root k to make the field F := F7(k), which will be of degree 3 over F7 and therefore of size 343. The ... 2answers 2k views ### show the rule between w, w^2, …w^{n-1} with w^n given w an nth root of unity Let w≠1 be an n-th root of unity, i.e., w^n-1=0. Show that 1+2w+3w^2+\dots+nw^{n-1}=-\frac n{1-w}. My question is how to relate w, w^2, \dots,w^{n-1} with w^n? 1answer 1k views ### Fastest Way to Find order of element in Finite Fields? Two questions: I use Miller-Rabin to find a prime, p, close to an arbitrary input number (this is very fast). Then I use Floyd's cycle finding algorithm to find the order of a randomly chosen element ... 1answer 230 views ### Cyclotomic Polynomials Let E(n) denote an nth root of unity. (For convenience, we may take E(n) = \exp(\frac{2πi}{n}).) Prove that for any prime p and any natural number r, we have$$ \prod_{\substack{j\\ \gcd(p^r, ... 1answer 114 views ### Discriminant and roots of$ x^{n^2} \pm (x-1)^{n^2}$? When considering the polynomials$x^{n^2} \pm (x-1)^{n^2}$($n$integer > 1 ) i noticed some things that appeared weird to me. Discriminant($x^{n^2} + (x-1)^{n^2}) = (n^2)^{n^2}$. Discriminant($x^{n^...
Determine all eigenvalues of the matrix $$A=\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}$$ and then determine a base for each eigenspace. It's easy to compute \$\chi_A(z)...