All Questions
Tagged with primitive-roots field-theory
21
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How do I prove that the primitive element of a field extension are this way.
I'm doing an introductory course of field theory and there is one excercise that as easy as it seems it bring me on my nerves. It states:
Let $α_1, \dots, α_n ∈ \mathbb{C}$ be the roots of an ...
- 1
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2
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Counting primitive elements in a finite field extension
I want to find the no. of elements $\alpha \in \mathbb{F}_{3^5}$ so that $\mathbb{F}_{3}(\alpha) = \mathbb{F}_{3^5}$(minimal polynomial of $\alpha$ is of degree 5). I know such things do exist but how ...
- 3,623
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Vanishing sums of integral linear combinations of roots of unity
Let $\{ \xi^{i} \}_{i=1}^{n}$ be $n$-th roots of unity for some positive integer $n$. It is well known that if $n$ is a prime integer, there will be $n-1$ primitive $n$-th roots of unity which are ...
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A question related to Primitive 2n - roots of unity
Consider the following question asked in an assignment worksheet which I am solving by myself.
If n is an odd integer such that K contains a primitive nth root of unity and $ char
K \neq 2 $, then K ...
user775699
1
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1
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Primitive roots of a finite field
I understand the definition of a primitive root of integers; however, I am quite confused trying to find the primitive roots of $\frac{\mathbb{Z}/3\mathbb{Z}[x]}{(x^3-x+\overline{1})}$. I know that $\...
- 466
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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
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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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1
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Minimal extension field of $\mathbb{F}_2$ such that
Find the minimal extension field of $\mathbb{F}_2$ such that this extension contains an element of order $21$?
Attempt: I know that such an extension of $\mathbb{F}_2$ is like $\mathbb{F}_{2^s}$ and $...
- 166
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1
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A general type of generator of the multiplicative group of a finite field
Let $p>2$ be a prime number and $\alpha \in \overline{\mathbb{F}_p}$. It generates a finite field $\mathbb{F}_p(\alpha)$. Is there some $u \in \mathbb{F}_p$ such that $ \alpha + u$ is a generator ...
- 165k
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Show that group ring QG is isomorphic to Q(e3)xQ.
Let G be a group of order 3 and e3 - primitive root of unity of order 3.
Show that group ring QG is isomorphic to Q(e3)xQ (Carthesian product).
Earlier I have shown that group ring Q(e3)G is ...
- 904
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357
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Why are the Galois groups that correspond to extensions which adjoin primitive roots of unity given by the group of units mod n
Considering all the following in the context of Galois theory.
I believe, given say the primitive $9^{th}$ root of unity, that this will have as its minimum polynomial , the cyclotomic polynomial
$\...
- 2,815
2
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2
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254
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Prove that a primitive $q$-th root of unity is in the algebraic closure of $\Bbb F_p$
Let $p$ and $q$ be odd primes. Let $\Omega$ be the algebraic closure of $\Bbb F_p$. Let $\omega$ be a primitive $q$-th root of unity. Show that $\omega \in \Omega$.
How do I show that? Please help me ...
- 4,862
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Primitive $p$-th root of unity with characteristic $p$
I struggle on this since two days, and still found no answer. My course states the following:
If the characteristic of $K \neq p$, then they are exactly $p-1$ different $p$-th roots of unity in the ...
- 21
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Show $2+\alpha$ is a primitive root of $\mathbb{F}_{25}$.
Suppose $\alpha \in \mathbb{F}_{25}$ is an element with $\alpha^2 = 2$, I need to prove that $2+\alpha \in \mathbb{F}_{25}$ is a primitive root (that is: a generator of the cyclic group $\mathbb{F}_{...
- 1,659
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Galois theory question
Let α be a complex primitive 43rd root of 1. Prove that there is an extension field F of the rational numbers such that $[F(\alpha): F] = 14$. I don't know how to start, can someone help me out?
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