Polynomial Rings: Finding 8 Elements with r^2=r

In summary, a polynomial ring is a mathematical structure that consists of polynomials with coefficients in a given ring. Finding elements with r^2=r in a polynomial ring is important for solving equations and understanding the properties of the ring. This can be done using techniques such as substitution, simplification, and algebraic methods. Polynomial rings have various practical applications in fields such as mathematics, physics, computer science, and engineering. One example of a polynomial ring where r^2=r is the ring of polynomials with coefficients in the field of real numbers.
  • #1
jgens
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Homework Statement



Find eight elements [itex]r \in \mathbb{Q}[x]/(x^4-16)[/itex] such that [itex]r^2=r[/itex].

Homework Equations



N/A

The Attempt at a Solution



The elements [itex]0+(x^4-16)[/itex] and [itex]1+(x^4-16)[/itex] clearly satisfy the desired properties, but I still need six more elements. Can anyone help me figure out a technique for finding a few more elements?

Thanks!
 
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  • #2
You can always brute force it. An arbitrary element in your ring has the form [itex][ax^3+bx^2+cx+d][/itex]. Square it and see when the relation is satisfied.
 

FAQ: Polynomial Rings: Finding 8 Elements with r^2=r

What is a polynomial ring?

A polynomial ring is a mathematical structure in abstract algebra that consists of polynomials with coefficients in a given ring. It is denoted by R[x], where R is a ring and x is an indeterminate or variable.

What is the purpose of finding 8 elements with r^2=r in a polynomial ring?

Finding elements with r^2=r in a polynomial ring is important for solving equations and understanding the properties of the ring. It helps in determining the structure and behavior of the ring, such as whether it is a field, a domain, or an ideal.

How do you find 8 elements with r^2=r in a polynomial ring?

To find 8 elements with r^2=r in a polynomial ring, you can use techniques such as substitution and simplification, or solving equations using algebraic methods. You can also use properties of the ring, such as commutativity and associativity, to manipulate the elements and find the desired solution.

What are some practical applications of polynomial rings?

Polynomial rings have various applications in mathematics, physics, computer science, and engineering. They are used in cryptography, coding theory, signal processing, and geometric modeling. They also have applications in fields such as economics, biology, and chemistry.

Can you give an example of a polynomial ring where r^2=r?

One example of a polynomial ring where r^2=r is the ring of polynomials with coefficients in the field of real numbers. In this case, r can be any real number, and r^2=r holds true for elements such as 0, 1, -1, and fractions like 1/2 and -3/4.

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