All Questions
Tagged with primitive-roots extension-field
12
questions
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37
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Inclusion of field extensions $\mathbb{Q}(\omega_p)$ and $\mathbb{Q}(\omega_{2p})$
Could you help me clear up a confusion? Either my following reasoning is wrong, or I should be able to show that $\omega_{2p} \in \mathbb{Q}(\omega_p)$, which does not seem correct.
Degree of Field ...
- 131
2
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0
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42
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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
2
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1
answer
48
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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
3
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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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45
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How to Simplify $\mathbb{Q}(\zeta_4,\zeta_8 , \zeta_{12},\zeta_6 )$
If $mdc(m,n)=1$ then $\mathbb{Q}(\zeta_m,\zeta_n )=\mathbb{Q}(\zeta_{mn} )$,but what if the degree of the roots are divisors and multiples of each other?
I guess that $\mathbb{Q}(\zeta_4,\zeta_6, \...
- 1,403
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36
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Subfields of Galois Extensions and association with Galois Groups
Let $\mathbb{K}\subseteq \mathbb{L}$ be a Galois extension with order $n$. If $p$ is a prime divisor of $n$, show that exists a subfield $\mathbb{M}$ of $\mathbb{L}$ such that $[\mathbb{L},\mathbb{M}]...
- 1,403
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2k
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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
1
vote
1
answer
292
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degree of extension of Q with primitive roots of unity
Question
Show that $[\mathbb{Q}(\zeta+\zeta^{-1}):\mathbb{Q}]=\phi(n)/2$, where $\zeta$ is a primitive $n$-th root of unity.
Attempt
Let $\sigma: z\mapsto \bar{z}$,$\sigma(\zeta)=\zeta^{-1}$.Since $...
- 5,153
2
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304
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Primitive elements of finite fields
Let $p$ be a prime number and $q=p^n$ for some positive integer $n$. $F_q[x]$ is the polynomial ring with coefficients in $F_q$. For any $M(x)\in F_q[x]$, define $\mathcal{R}(M(x))\subset F_q[x]$ to ...
- 1,082
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1
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34
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Generators of $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ - what is $\tau(\zeta_n)$?
I am trying to understand the structure of $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q})\simeq Z_n^*$ for $n \in \mathbb{Z}$ - it would be great if someone could me understand the generators in this group so ...
- 81
2
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2
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343
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Irreducibility of $X^5-7$ over $\mathbb{Q}(\sqrt[7]{2})[X]$ and degree of spitting field
I have worked through these two questions but am unsure if I got the right idea, please may you help me?
Prove that $X^5-7$ is irreducible over $\mathbb{Q}(\sqrt[7]{2})[X]$
Can we say that $f(X)=X^5-...
- 713
1
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1
answer
511
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Number of fields between $\mathbb{Q}$ and $\mathbb{Q}(\zeta_n)$
If $\zeta_n$ is the $n$-th primitive root of unity then $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \simeq Z_n^*$ due to the following map
$$\tau(\zeta_n)=\zeta^n$$
I was wondering if we could use this and ...
- 713