# Abstract algebra questions and answers

Recent questions in Abstract algebra
Paula Good 2022-03-25 Answered

2022-03-21

### Calculate e and f for

Caroline Carey 2022-03-18 Answered

### Applications of the concept of homomorphismWhat 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 $f\left({X}^{2}+{Y}^{2}\right)=f\left(0\right),f\left({X}^{2}\right)+f\left({Y}^{2}\right)=0,{f\left(X\right)}^{2}+{f\left(Y\right)}^{2}=0,{x}^{2}+{y}^{2}=0$.

Juliet Jackson 2022-03-17 Answered

### Are the following true or false?1) $\mathrm{\forall }n\in \mathbb{N}:\left[\mathbb{Q}\left(\mathrm{exp}\left(2\pi \frac{i}{n}\right):\mathbb{Q}\right]=n-1$2) Let $f\in \mathbb{Q}\left[X\right]$ be irreducible, $deg\left(f\right)=n$. Then: $|Gal\left(f\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 commutative4) Let $\frac{L}{K}$ be a finite field extension and $|Gal\left(\frac{L}{K}\right)|=1$. Then $\frac{L}{K}$ is normal.

Tianna Costa 2022-03-17 Answered

### 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?

Jerimiah Boone 2022-03-14 Answered

### Why the ring ${S}^{-1}R$ is Noetherian if S is multiplicative?

Umaiza Hutton 2022-03-03 Answered

### In the group $GL\left(2,{\mathbb{Z}}_{7}\right)$, inverse of $A-\left(\begin{array}{cc}4& 5\\ 6& 3\end{array}\right)$ is ?

Ingrid Senior 2022-03-01 Answered

### A question on Hamilton QuaternionsHow does one prove that ring of Hamilton Quaternions with coefficients coming from the field $\frac{\mathbb{Z}}{p}\mathbb{Z}$ is not a divison ring.

Junaid Ayala 2022-03-01 Answered

### If G is a finite group with $|G|<180$ and G has subgroups of orders 10, 18 and 30 then the order of G is:1) 602) 903) 304) 80

Erik Sears 2022-03-01 Answered