All Questions
Tagged with primitive-roots abstract-algebra
53
questions
3
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answer
58
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Two primitive roots add to a third
Primitive roots are useful in analyzing the multiplicative structure of the numbers mod $p$, for $p$ a prime. I wondered if there was anything that could link the multiplicative and additive structure ...
- 4,764
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vote
4
answers
238
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How to find a primitive element for $\mathbb F_{11}$
This is a pretty dumb question considering this is the very first question in my exercises (on a section about finite fields). First, the the group of units $\mathbb F_{11}^*$ is $\mathbb{Z}/10\mathbb{...
- 840
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0
answers
53
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When does this Galois theorem "variation" holds?
Im reading about solvability by radicals in different books, more concretely in Fields and Galois Theory by Patrick Morandi and Introduction to abstract algebra by Benjamin Fine, etc. In the second ...
- 23
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2
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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
answer
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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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0
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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
vote
1
answer
668
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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
answer
272
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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
1
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1
answer
45
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Uniqueness of automorphisms on root
The following is from A. Białynicki-Birula, "Algebra" (the translation is mine).
Example 3. Let $p$ be a prime number. Let us consider the splitting field $L$ of the polynomial $\frac{x^p - ...
- 302
0
votes
2
answers
37
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Uniqueness of values of automorphisms on a primitive root
Let $L$ be a field and let $\zeta \in L$ be a primitive root of $n$-th degree. Let $\alpha$ and $\beta$ be automorphisms belonging to $Aut(L)$. Is it possible that $\alpha(\zeta) \ne \beta(\zeta)$? It ...
- 302
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1
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75
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Relationships between the Orders of an element in a Cyclic group (multiplicative and additive) Relating to prime numbers
Let p be a prime. Then, we know that $U(\mathbb{F}_p) \simeq \mathbb{Z}/(p-1)\mathbb{Z}$, where $U(\mathbb{F}_p)$ is the group of units of the field $\mathbb{F}_p$. They both have order $p-1$.
Given ...
- 424
4
votes
0
answers
156
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Is there a deeper reason for the classification of moduli in which a primitive root exists?
The primitive root theorem classifies the set of moduli for which a primitive root exists as $$1,2,4,p^k,2p^k$$ where $p$ is an odd prime and $k$ is a positive integer.
I have worked through a proof ...
- 3,415
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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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0
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34
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Determine degree min. polynomial
I need a check on the following question
Let $\alpha$ a primitive element of $\mathbb{F}_{2^n}$. Determine the degree of the minimal polynomial over $\mathbb{F}_2$. What can you say about the ...
- 166
0
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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 ...
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