Solving Equiv. Class Problems in Q[t]: Add, Mult, Zero Divisors

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In summary, the equivalence relation ~ is a well-defined addition/multiplication of equivalence classes when f(t) - g(t) is a multiple of t^2-5.
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hopsh
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In Q[t], define the equivalence relation ~ as f(t) ~ g(t) precisely when f(t) - g(t) is a multiple of t^2 - 5. We define the addition and multiplication of equivalence classes as [f(t)] + [g(t)] = [f(t) + g(t)] and [f(t)] * [g(t)] = [f(t) * g(t)]
(Assume: ~ is an equivalence relation, Addition/Mutliplication of equivalence classes is well-defined, and every equivalence class contains exactly one element of the form a + bt, where a, b in Q)

a) Find a, b in Q such that [3t^3 - 5t^2 + 8t - 9] = [a + bt]
b) Find a, b in Q such that [2t + 7] * [7t + 11] = [a + bt]
c) Find two equivalence classes whose square is equal to [5]
d) Find a, b in Q such that [a + bt]^2 + [-2][a + bt] = [19] (Two possible answers)
e) Find a, b in Q such that [2 - t] * [a + bt] = [1]
f) Which equivalence classes are zero divisors?
g) Which equivalence classes have multiplicative inverses?
h) How many equivalence classes are there whose square is equal to [6] ?

Note: * means multiplication


Now, I think I found the following solutions. Can someone verifty these and help solve the remaining parts?

a) [35t - 54]
b) [71t + 147]
c) ??
d) ??
e) ??
f) [0] = [t^2 - 5] (are these the only zero divisor classes??)
g) ??
h) ??
 
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  • #2
What does [t] behave like? This should give you a better feel for this field (whose elements are the equivalence classes).
 
  • #3
I'm lost with this problem and I have a handful of these to do. If I had an example of how to solve one of these I'm confident I can figure out the rest (I learn by example and since there aren't any with the materials I have (not even any odd solutions in the back!) I'm really desparate). Please, is there any way you could post the solutions to this one with some intermediate explanations.
 
  • #4
If you think about what I said for a few minutes, you'll get all the answers very easily, along with a deeper understanding. No one is going to do the problems for you, and you should be grateful for that.
 
  • #5
How did you get the answers to the parts you have done? If you explain that then we'll have a better idea of what you understand.
 
  • #6
Maybe I should have been more clear. [t^2-5]=[t]^2-[5]=[0], so in a sense, [t] behaves just like sqrt(5). See how far you can push this analogy.
 

FAQ: Solving Equiv. Class Problems in Q[t]: Add, Mult, Zero Divisors

What are equivalence classes in Q[t] and why are they important in problem-solving?

Equivalence classes in Q[t] are sets of polynomials that are considered "equal" under a certain set of rules. These rules are determined by the type of problem being solved. Equivalence classes are important in problem-solving because they allow us to simplify complex equations and focus on finding solutions within a smaller set of polynomials.

How do you add two polynomials in Q[t]?

To add two polynomials in Q[t], we follow the same rules as adding any two polynomials. We combine like terms and arrange the resulting polynomial in descending order of degree. If there are any coefficients that are equivalent in the resulting polynomial, we can simplify by combining them into one term.

What is the process for multiplying two polynomials in Q[t]?

Multiplying two polynomials in Q[t] is similar to multiplying any two polynomials. We use the distributive property to multiply each term of the first polynomial by each term of the second polynomial. We then combine like terms and arrange the resulting polynomial in descending order of degree. If there are any coefficients that are equivalent, we can simplify by combining them into one term.

What are zero divisors in Q[t] and how do they impact solving problems?

Zero divisors in Q[t] are polynomials that, when multiplied by another polynomial, result in the zero polynomial. In other words, they are "divisible" by other polynomials. Zero divisors can impact solving problems because they can create more complex equations and can lead to multiple solutions or no solution at all.

Can you provide an example of solving an equivalence class problem in Q[t]?

Sure, let's say we are trying to solve the equation x + 2 = 0 in Q[t]. We can create two equivalence classes: [x] and [0]. Since we want to find a polynomial that is equivalent to 0, we can choose any polynomial in the [0] equivalence class, such as x^2 - x. We can then add this polynomial to both sides of the equation, resulting in x + 2 + (x^2 - x) = 0 + (x^2 - x). Simplifying, we get x^2 + x + 2 = x^2 - x. This equation has no solutions, so the [x] equivalence class is empty and there is no polynomial equivalent to x that satisfies the original equation.

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