Recent questions in Abstract algebra

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siliciooy0j 2022-03-31

Prove if $F\left(\sqrt[n]{a}\right)$ is unramified or totally ramified in certain conditions

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Rex Maxwell 2022-03-30

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Aleah Choi 2022-03-29

I have to prove that if P is a R-module , P is projective $\iff$ there is a family $\left\{{x}_{i}\right\}$ in P and morphisms ${f}_{i}:P\to R$ such that for all $x\in P$

$x=\sum _{i\in I}{f}_{i}\left(x\right){x}_{i}$

where for each$x\in P,\text{}{f}_{i}\left(x\right)=0$ for almost all $i\in I$ .

where for each

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Jasper Dougherty 2022-03-29

Proving that the set of units of a ring is a cyclic group of order 4

The set of units of $\frac{\mathbb{Z}}{10}\mathbb{Z}$ is $\{\stackrel{\u2015}{1},\stackrel{\u2015}{3},\stackrel{\u2015}{7},\stackrel{\u2015}{9}\}$, how can I show that this group is cyclic?

My guess is that we need to show that the group can be generated by some element in the set, do I need to show that powers of some element can generate all elements in the other congruence classes?

For example ${7}^{2}=49\equiv 9\pm \mathrm{mod}10$, i.e. using 7 we can generate an element in the congruence class of 9, but can not generate 29 for example from any power of 7, so is it sufficent to say that an element is a generator if it generates at least one element in all other congruence classes?

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r1fa8dy5 2022-03-28

Proving the generator of

$A=\{154a+210b:a,b\in \mathbb{Z}\}$ is $(154,\text{}210)$

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Hugh Soto 2022-03-28

Prove that $n\mid \varphi ({a}^{n}-1)$. Any natural solution that uses $Aut\left(G\right)$

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Marzadri9lyy 2022-03-28

Prove that $\mathbb{Z}+\left(3x\right)$ is a subring of $\mathbb{Z}\left[x\right]$ and there is no surjective homomorphism from $\mathbb{Z}\left[x\right]\to \mathbb{Z}+\left(3x\right)$

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anadyrskia0g5 2022-03-27

Definition of polynomial ring

Given a ring R, the polynomial ring is defined as

$R\left[x\right]\phantom{\rule{0.222em}{0ex}}=\{\sum _{k=0}^{n}{a}_{k}{x}^{k}:n\ge 0,\text{}{a}_{k}\in R\text{}\text{for}\text{}k\in \{0,1,\cdots ,n\}\}.$

However, it is not usually specified what x is. In order for multiplication to make sense, I guess it has to be an element in R at least. But is R[x] the set of all functions $P:R\to R,\text{given by}\text{}x\mapsto \sum _{k=0}^{n}{a}_{k}{x}^{k}$, or the set of those functions evaluated at x?

Sometimes R is required to be commutative. Does that make any difference for R[x]?

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Asher Olsen 2022-03-25

How do I factorize ${x}^{6}-1$ over GF(3)? I know that the result is $(x+1)}^{3}{(x+2)}^{3$ , but I'm unable to compute it myself.

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Paula Good 2022-03-25

How can I show these polynomials are not co'?

$x,\text{}x-1$ and $x+1$ in the ring ${\mathbb{Z}}_{6}\left[x\right]$

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siliciooy0j 2022-03-24

Let k be a field, V a finite-dimensional k-vectorspace and $M\in End\left(V\right)$ . How can I determine Z, the centralizer of $M\otimes M$ in $End\left(V\right)\otimes End\left(V\right)$ ?

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Pelizzolaf40 2022-03-19

Calculate e and f for ${\mathbb{Q}}_{2}(\sqrt{3},\text{}\sqrt{2})$

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Caroline Carey 2022-03-18

Are all algebraic integers with absolute value 1 roots of unity?

If we have an algebraic number α with (complex) absolute value 1, it does not follow that α is a root of unity (i.e., that$\alpha n=1$ for some n). For example, $(\frac{3}{5}+\frac{4}{5i})$ is not a root of unity.But if we assume that $\alpha$ is an algebraic integer with absolute value 1, does it follow that $\alpha$ is a root of unity?

If we have an algebraic number α with (complex) absolute value 1, it does not follow that α is a root of unity (i.e., that

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fun9vk 2022-03-18

Applications of the concept of homomorphism

What are some interesting applications of the concept of homomorphism?

Example: If there is a homorphism from a ring R to a ring r then a solution to a polynomial equation in R gives rise to a solution in r. e.g. if$f:R\to r$ and ${X}^{2}+{Y}^{2}=0$ then $f({X}^{2}+{Y}^{2})=f\left(0\right),f\left({X}^{2}\right)+f\left({Y}^{2}\right)=0,{f\left(X\right)}^{2}+{f\left(Y\right)}^{2}=0,{x}^{2}+{y}^{2}=0$ .

What are some interesting applications of the concept of homomorphism?

Example: If there is a homorphism from a ring R to a ring r then a solution to a polynomial equation in R gives rise to a solution in r. e.g. if

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Juliet Jackson 2022-03-17

Are the following true or false?

1)$\mathrm{\forall}n\in \mathbb{N}:[\mathbb{Q}(\mathrm{exp}\left(2\pi \frac{i}{n}\right):\mathbb{Q}]=n-1$

2) Let$f\in \mathbb{Q}\left[X\right]$ be irreducible, $deg\left(f\right)=n$ . Then: $\left|Gal\left(f\right)\right|=n$

3) Let$\frac{K\left(a\right\}}{K}$ be an algebraic field extension. Then $Gal\left(\frac{K\left(a\right)}{K}\right)$ is commutative

4) Let$\frac{L}{K}$ be a finite field extension and $\left|Gal\left(\frac{L}{K}\right)\right|=1$ . Then $\frac{L}{K}$ is normal.

1)

2) Let

3) Let

4) Let

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Tianna Costa 2022-03-17

Can we find a binary operation s.t. the map $\varphi \left(n\right)=n+1$ becomes an isomorphism from $\u27e8B\mathbf{Z},\cdot \u27e9$ onto $\u27e8B\mathbf{Z},.\u27e9$ ?

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pinka1hf 2022-03-14

When does the product of two polynomials $={x}^{k}$ ?

Suppose f and g are are two polynomials with complex coefficents (i.e$f,g\in \mathbb{C}\left[x\right]$ ). Let m be the order of f and let n be the order of g.

Are there some general conditions where$fg=\alpha {x}^{n+m}$ for some non-zero $\alpha \in \mathbb{C}$ .

Suppose f and g are are two polynomials with complex coefficents (i.e

Are there some general conditions where

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Jerimiah Boone 2022-03-14

Why are the only division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?

Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?

Here a division algebra is an associative algebra where every nonzero number is invertible (like a field, but without assuming commutativity of multiplication).

Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?

Here a division algebra is an associative algebra where every nonzero number is invertible (like a field, but without assuming commutativity of multiplication).

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Thaddeus Nolan 2022-03-14

Why can't the Polynomial Ring be a Field?

I'm currently studying Polynomial Rings, but I can't figure out why they are Rings, not Fields. In the definition of a Field, a Set builds a Commutative Group with Addition and Multiplication. This implies an inverse multiple for every Element in the Set.

The book doesn't elaborate on this, however. I don't understand why a Polynomial Ring couldn't have an inverse multiplicative for every element (at least in the Whole numbers, and it's already given that it has a neutral element). Could somebody please explain why this can't be so?

I'm currently studying Polynomial Rings, but I can't figure out why they are Rings, not Fields. In the definition of a Field, a Set builds a Commutative Group with Addition and Multiplication. This implies an inverse multiple for every Element in the Set.

The book doesn't elaborate on this, however. I don't understand why a Polynomial Ring couldn't have an inverse multiplicative for every element (at least in the Whole numbers, and it's already given that it has a neutral element). Could somebody please explain why this can't be so?

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