Polynomials in Z6[x]: Find & Explain Deg 0 Product

  • Thread starter sarah77
  • Start date
  • Tags
    Polynomials
In summary, In Z6[x] and Z7[x], the only variable can be x. The product of two polynomials in Z6[x] must have a degree of 6 to be a degree of 0. In Z7[x], there are no zero divisors, so the product of two polynomials cannot have a degree of 0.
  • #1
sarah77
27
0

Homework Statement



Find two polynomials, each of degree 2, in Z6[x] whose product has degree 0. Can you repeat the same in Z7[x]? Explain.

Homework Equations



In Z6[x] and Z7[x] can the only variable be x?

The Attempt at a Solution



I know Z6 consists of {0,1,2,3,4,5} and Z7: {0,1,2,3,4,5,6}; and I have tried (x2+5)(x2-3) and others but I get degree of 4, and it must be degree of 6 for it to be degree 0 in Z6..Please help, am I confused on how to solve this.
 
Physics news on Phys.org
  • #2
What's 3x^2 times 2x^2 in Z6?
 
  • #3
6x^4, so 0x^4...the polynomial is of degree 4, but since the coefficient is 0, the product would have a degree of 0?
 
  • #4
sarah77 said:
6x^4, so 0x^4...the polynomial is of degree 4, but since the coefficient is 0, the product would have a degree of 0?

Sure. Now why can't that happen in Z7?
 
  • #5
Since it is a prime number, no two elements in Z7 can be multiplied to obtain a number divisible by 7.
 
  • #6
sarah77 said:
Since it is a prime number, no two elements in Z7 can be multiplied to obtain a number divisible by 7.

Exactly. There are no zero divisors in Z7. There are in Z6.
 
  • #7
Thank you, that makes sense!
 

FAQ: Polynomials in Z6[x]: Find & Explain Deg 0 Product

1. What is a polynomial in Z6[x]?

A polynomial in Z6[x] is an expression consisting of constants, variables, and coefficients, using the operations of addition, subtraction, and multiplication. In this case, the coefficients are restricted to the integers modulo 6 (Z6), and the variable is typically denoted by x.

2. What does it mean to find a polynomial of degree 0 in Z6[x]?

Finding a polynomial of degree 0 in Z6[x] means finding a constant term, or a polynomial with no variables. In other words, the polynomial would be of the form c, where c is an integer modulo 6.

3. How do you find the degree of a polynomial in Z6[x]?

The degree of a polynomial in Z6[x] is the highest power of the variable x in the polynomial. In other words, it is the highest exponent of x that appears in the polynomial. For example, the polynomial 2x3 + 4x2 + 6 has a degree of 3.

4. What is the product of two polynomials in Z6[x]?

The product of two polynomials in Z6[x] is another polynomial in Z6[x] obtained by multiplying each term of one polynomial by each term of the other polynomial. The resulting polynomial may have a higher degree than the original polynomials, but its coefficients will still be integers modulo 6.

5. How do you explain the deg 0 product of two polynomials in Z6[x]?

The deg 0 product of two polynomials in Z6[x] is the result of multiplying two polynomials of degree 0. This means that each polynomial only has a constant term, and when multiplied together, the resulting polynomial will also have a constant term. In other words, the deg 0 product of two polynomials in Z6[x] is simply the product of their constant terms.

Similar threads

What are the cyclic subgroups in Z6 x Z3?
Replies
5
Views
7K
Can Zero Divisors in Polynomial Rings Be Characterized?
Replies
13
Views
2K
Finding Integer Solutions to Polynomial Equations: Can it be Done Easily?
Replies
9
Views
2K
Given the zhegalkin polynomial, find the boolean function
Replies
1
Views
1K
Finding the minimal polynomial of primitive 15th root of 1
Replies
4
Views
3K
Matrix over Z7 needs revision please
Replies
2
Views
1K
Find the basis so that the matrix will be diagonal
Replies
14
Views
2K
Solve Newton Interpolating Polynomial for Error
Replies
6
Views
1K
How to convince myself that I can take n=1 here?
Replies
1
Views
815
Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##
Replies
7
Views
1K

Hot Threads

Recent Insights

Back
Top