All Questions
Tagged with primitive-roots roots-of-unity
39
questions
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73
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Characterizing generators for the multiplicative group of a finite field.
Fix a finite field $\mathbb{F}_p$ and consider its multiplicative group $\mathbb{F}_p^\times$, which we know is cyclic. Is there an general way to characterize this group's generators (the primitive $(...
- 990
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1
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186
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Understanding Neukirch´s proof
I´m studying algebraic number theory from Neukirch´s book. I´m reading the Proposition 10.2 which says:
A $\mathbb{Z}$-basis of the ring $O$ of integers of $\mathbb{Q}(\zeta)$ is given by $1, \zeta, \...
- 133
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110
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Why it is sufficient to look at prime divisor of $p-1$ when finding generators of $\mathbb{Z}_p^*$?
Let's say that I want to find the generators of $\mathbb{Z}_p^*$, where $p$ is a prime number. I found the following necessary and sufficient condition:
An element $x \in \mathbb{Z}_p^*$ is a ...
- 321
1
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120
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A Primitive element in Finite Fields
My question is about the roots of unity in finite fields. It goes like this:
Suppose we have two primes p and q, both greater than 3, which satisfy $q|(p-1)$. Then there exists a $q$-the root of unity ...
- 301
2
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Quadratic number fields that contain primitive root of unity
Find all quadratic fields $\mathbb{Q}[\sqrt{d}]$ that contain some $p$-th primitive root of unity, where $p>2$ is a prime.
Now, my reasoning was: if $\mathbb{Q}[\sqrt{d}]$ contains one $p$-th root ...
- 816
1
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1
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71
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Finding a counter-example for Gaussian-periods for non-primes
I need to give a counter-example against the following theorem:
Suppose $H \subset \operatorname{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ is a subgroup. Then we have $\mathbb{Q}(\zeta_n)^H = \mathbb{Q}(\...
- 1,659
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53
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Determine if a $4$-th root of unity is contained in $\mathbb{F}_9$
I have two questions, one of those is the same here, but I'd like to use another argument and I need a check!
The text is:
i) Is it true that a primitive 3-th root of the unit over $\mathbb{F}_3$ is ...
- 185
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81
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Smallest field and root of unity
I'm trying to solve the following:
Let $K$ be the smallest field, with characteristic $2$ such that it contains a $15$-primitive root of the unit. Find its cardinality and a primitive element of ...
- 166
1
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3
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159
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what is the value of $\binom{n}{1}+\binom{n}{4}+\binom{n}{7}+\binom{n}{10}+\binom{n}{13}+\dots$
what is the value of $$\binom{n}{1}+\binom{n}{4}+\binom{n}{7}+\binom{n}{10}+\binom{n}{13}+\dots$$ in the form of number,
cos, sin
attempts : I can calculate the value of $$\binom{n}{0}+\binom{n}{...
1
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1
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381
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Primitive roots of the unity in $\mathbb C$
Let $\omega$ be a primitive $n-$th root of unity.
(i) Show that its powers $\omega^k$, for $k ∈ {1, \ldots, n}$, are all different;
(ii) Deduce that they are precisely all the $n-$th roots of unity.
...
user720013
0
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1
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131
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Is $\bar{x}$ a primitive element in $\mathbb{F}_p[x]/(P(x))$
Let $\mathbb{F}_p$ be the finite field of $p$ elements (where $p$ is a prime) and let $P(x)\in \mathbb{F}_p[x]$ be an irreducible polynomial. I have to prove that $\bar{x}$ is a primitive element of $\...
- 1,016
3
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1
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220
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Why is $[\mathbb{Q}(\zeta):\mathbb{Q}] = 8$ and not $14$? (Where $\zeta$ is a primitive $15^{th}$ root of unity)
I have a field extension $\mathbb{Q}(\zeta)/\mathbb{Q}$, where $\zeta$ is a primitive $15^{th}$ root of unity.
So, since $x^{15}-1 = \phi_{1}(x)\phi_{3}(x)\phi_{5}(x)\phi_{15}(x)$, where $\phi_{n}(x)$...
- 443
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32
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Let $w \in G_{63}$ be a primitive root of unity. Find all $n\geq 6$ such that $\sum_{k=6}^n w^{35k} = 0$ and $w^{12n} = w^{15}$
This is probably totally wrong.
We know that $5$ and $63$ are relatively prime, therefore $w^5 \in G_{63}$ primitive (we'll suppose $w= w^5$ without loss of generality).
We also know that $(w^7)^9 = ...
- 781
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1
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155
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Rational coefficients of the prime roots of unity (basis set in $\mathbb{Q}$).
It is well known that the set of prime roots of unity $S = \{\zeta, \zeta^2, \zeta^3, ..., \zeta^{p-1}\}$ form a basis in $\mathbb{Q}$ (where $\zeta = e^{\frac {i2\pi}{p}}$, $p \in \mathbb{N}, p $ is ...
- 3
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Let $w \in G_{15}$ be a primitive root. Find every $n \in \mathbb{N}$ such that $\sum_{i=2}^{n-1} w^{3i} = 0$
We can first rewrite the series in a useful form,
$$\sum_{i=2}^{n-1} w^{3i} = \bigg( \sum_{i=0}^{n-1} w^{3i} \bigg) - w^3 - 1 $$
But since $w$ is primitive, we can apply the geometric series formula,...
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