Im

scrimaeua
2022-02-12
Answered

Order of field extension

Im

Im

You can still ask an expert for help

meldElafellrbo

Answered 2022-02-13
Author has **13** answers

Let $f={x}^{3}+a{x}^{2}+bx+c$ be an irreducible polynomial over $\mathbb{Z}}_{11$ .

The extension field$E={\mathbb{Z}}_{11}\frac{x}{\u27e8f\u27e9}$ contains a zero of f, namely the residue class $\alpha =\stackrel{\u2015}{x}+\u27e8f\u27e9$ , This gives ${\alpha}^{3}+a{\alpha}^{2}+b\alpha +c=,\text{so}\text{}{\alpha}^{3}=-a{\alpha}^{2}-b\alpha -c$ , and the extension field is $E=\{u{\alpha}^{2}+v\alpha +w\mid u,v,w\in {\mathbb{Z}}_{11}\}$ with degree $[E:{\mathbb{Z}}_{11}]=3$ . In particular, if f is primitive, the powers of $\alpha$ are exactly the nonzero elements of E.

NB:$f\left(\stackrel{\u2015}{x}\right)=f(x+\u27e8f\u27e9)=f\left(x\right)+\u27e8f\u27e9=\u27e8f\u27e9=\stackrel{\u2015}{0}$ .

The extension field

NB:

asked 2020-11-20

Prove that in any group, an element and its inverse have the same order.

asked 2021-09-27

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic.

asked 2021-06-23

Determine whether the statement is true or false. If the last row of the reduced row echelon form of the augmented matrix of a system of linear equations has only one nonzero entry, then the system is inconsistent.

asked 2021-09-25

Find the cross product

asked 2021-06-15

Graph f and g in the same rectangular coordinate system. Use transformations of the graph off to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function’s domain and range.

asked 2021-05-08

In Exercises 6 through 9, write the set in the form

6.

7.

8.

9.

asked 2022-06-08

Is every finite ring a matrix algebra over a commutative ring?

- Can every finite ring $R$ be written as a subring of ${\text{Mat}}_{n\times n}A$ for some commutative ring $A$?

- If not, then what is/are the smallest ring(s) that cannot be?

- Can every finite ring $R$ be written as a subring of ${\text{Mat}}_{n\times n}A$ for some commutative ring $A$?

- If not, then what is/are the smallest ring(s) that cannot be?