 # Abstract algebra questions and answers

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

### How can I show these polynomials are not co'? and $x+1$ in the ring ${\mathbb{Z}}_{6}\left[x\right]$ siliciooy0j 2022-03-24 Answered

2022-03-21

### Find all zero divisors of z_81 Pelizzolaf40 2022-03-19 Answered

### Calculate e and f for Caroline Carey 2022-03-18 Answered

### 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, $\left(\frac{3}{5}+\frac{4}{5i}\right)$ 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? fun9vk 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

### Can we find a binary operation s.t. the map $\varphi \left(n\right)=n+1$ becomes an isomorphism from $⟨B\mathbf{Z},\cdot ⟩$ onto $⟨B\mathbf{Z},.⟩$? pinka1hf 2022-03-14 Answered

### 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 $\alpha \in \mathbb{C}$. Thaddeus Nolan 2022-03-14 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 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). Kienastsrx 2022-03-12 Answered

### Why is $\mathbb{Q}\left(t,\sqrt{{t}^{3}-t}\right)$ not a purely transcendental extension of $\mathbb{Q}$? Vandinimo3 2022-03-09 Answered

### Without actually computing the orders, explain why the two elements in each of the following pairs of elements from Z30 must have the same order: {2, 28}, {8, 22}. Do the same for the following pairs of elements from U(15): {2, 8}, {7, 13}. ordanlarroque1xm 2022-03-08 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

### Write $\gamma =\left(124\right)\left(425\right)\left(64\right)$ as a product of disjoint cycles. Havlishkq 2022-02-28 Answered

### Compute the kernel for the given homomorphism $\varphi$.$\varphi :\mathbb{Z}\to {\mathbb{Z}}_{8}$ such that $phI\left(1\right)=6$.

Coming up with good abstract algebra examples is essential for those who are trying to come up with the answers to theoretical questions both in Engineering and Data Science disciplines. The college students will be able to discover abstract algebra questions and answers provided by our friendly experts that will help you to understand abstract algebra questions with various examples based on high-energy physics, cryptography, and the number theory. Remember that the trick is to use number sequences to generalize the set of various integers and transformations of functions equation problems at play. Don’t forget about the application of the Algebraic number theory studies as well.