Chapter 33, Cyclotomic Extensions Video Solutions, Contemporary Abstract Algebra

02:43

Problem 1

Determine the minimal polynomial for $\cos (\pi / 3)+i \sin (\pi / 3)$ over $Q$.

Aman Gupta

Numerade Educator

00:43

Problem 2

Factor $x^{12}-1$ as a product of irreducible polynomials over $Z$.

Sherrie Fenner

Numerade Educator

00:41

Problem 3

Factor $x^{8}-1$ as a product of irreducible polynomials over $Z_{2}, Z_{3}$, and $Z_{5}$.

Khanh Ha

Numerade Educator

03:04

Problem 4

For any $n>1$, prove that the sum of all the $n$ th roots of unity is 0 .

Farnood Ensan

Numerade Educator

01:21

Problem 5

For any $n>1$, prove that the product of the $n$ th roots of unity is $(-1)^{n+1}$

Kevin Harmer

Numerade Educator

03:08

Problem 6

Let $\omega$ be a primitive 12 th root of unity over $Q$. Find the minimal polynomial for $\omega^{4}$ over $Q$.

Chris Trentman

Numerade Educator

04:15

Problem 7

Let $F$ be a finite extension of $Q$. Prove that there are only a finite number of roots of unity in $F$.

Donald Albin

Numerade Educator

01:46

Problem 8

For any $n>1$, prove that the irreducible factorization over $Z$ of $x^{n-1}+x^{n-2}+\cdots+x+1$ is $\Pi \Phi_{d}(x)$, where the product runs over all positive divisors $d$ of $n$ greater than 1 .

Sanchit Gogia

Numerade Educator

02:57

Problem 9

If $2^{n}+1$ is prime for some $n \geq 1$, prove that $n$ is a power of 2 . (Primes of the form $2^{n}+1$ are called Fermat primes.)

Bryan Lynn

Numerade Educator

01:42

Problem 10

Prove that $\Phi_{n}(0)=1$ for all $n>1$.

Anna Jones

Numerade Educator

01:22

Problem 11

Prove that if a field contains the $n$ th roots of unity for $n$ odd, then it also contains the $2 n$ th roots of unity.

Sriram Soundarrajan

Numerade Educator

03:56

Problem 12

Let $m$ and $n$ be relatively prime positive integers. Prove that the splitting field of $x^{m n}-1$ over $Q$ is the same as the splitting field of $\left(x^{m}-1\right)\left(x^{n}-1\right)$ over $Q$

Sriparna Bhattacharjee

Numerade Educator

02:01

Problem 13

Prove that $\Phi_{2 n}(x)=\Phi_{n}(-x)$ for all odd integers $n>1$.

Mohamed Mohamed

Numerade Educator

01:44

Problem 14

Prove that if $p$ is a prime and $k$ is a positive integer, then $\Phi_{p^{k}}(x)=$ $\Phi_{p}\left(x^{p^{k-1}}\right)$. Use this to find $\Phi_{8}(x)$ and $\Phi_{27}(x)$.

Aymara Gallardo

Numerade Educator

02:05

Prove the assertion made in the proof of Theorem $33.5$ that there exists a series of subgroups $H_{0} \subset H_{1} \subset \cdots \subset H_{t}$ with $\left|H_{i+1}: H_{i}\right|=2$ for $i=0,1,2, \ldots, t-1$. (This exercise is referred to in this chapter.)

Varsha Aggarwal

Numerade Educator

08:41

Problem 16

Prove that $x^{9}-1$ and $x^{7}-1$ have isomorphic Galois groups over $Q$. (See Exercise 7 in Chapter 32 for the definition.)

Chris Trentman

Numerade Educator

04:02

Problem 17

Let $p$ be a prime that does not divide $n .$ Prove that $\Phi_{p n}(x)=$ $\Phi_{n}\left(x^{p}\right) / \Phi_{n}(x)$

Mengchun Cai

Numerade Educator

01:05

Problem 18

Prove that the Galois groups of $x^{10}-1$ and $x^{8}-1$ over $Q$ are not isomorphic.

Anthony Ramos

Numerade Educator

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Problem 19

Let $E$ be the splitting field of $x^{5}-1$ over $Q .$ Show that there is a unique field $K$ with the property that $Q \subset K \subset E$.Let $E$ be the splitting field of $x^{5}-1$ over $Q .$ Show that there is a unique field $K$ with the property that $Q \subset K \subset E$.

Victor Salazar

Numerade Educator

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