Rings and Fields - Write down the nine elements of F9

In summary, F9[i] is a set consisting of all numbers a+bi with a and b in F9, where i is a solution to the equation i^2 = -1. The nine elements of F9[i] can be written as 0, 1, 2, i, 1+i, 2+i, 2i, 1+2i, and 2+2i. It can be shown that every nonzero element of F9[i] has an inverse, making F9[i] a field.
  • #1
cooljosh2k2
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Rings and Fields - Write down the nine elements of F9

Homework Statement



In F9 = Z/3Z, there is no solution of the equation x^2 = −1, just as in R. So “invent”
a solution, call it 'i'. Then 'i' is a new “number” which satisfies i^2 = −1. Consider
the set F9 consisting of all numbers a+bi, with a,b in F9. Add and multiply these
numbers as though they were polynomials in 'i', except whenever you get i^2 replace
it by −1.
(i) Write down the nine elements of F9 .
(ii) Show that every nonzero element of F9 has an inverse, so that F9 is a
field.


The Attempt at a Solution



I know I am supposed to show you that I've tried the question if i want an answer. Believe me, i have tried it. I am just really confused by the wording of the question and am not really sure what they are looking for in part a. Once i get part a, I am pretty sure id be able to get part b on my own.
 
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  • #2


You are told that F9 = {a+bi | a,b in F9}. The first part is asking you to try out all the elements. So for example, 1+2i is an element.
The second part is asking you to show that for all a+bi in F9, there is some c+di such that (a+bi)(c+di) = 1. Try writing the inverse out using a and b in some way, then multiply it out to check if you are correct and get 1.
 

FAQ: Rings and Fields - Write down the nine elements of F9

1. What is the difference between a ring and a field?

A ring is a mathematical structure that has two operations, addition and multiplication, and follows certain rules such as commutativity and distributivity. A field is a more specific type of ring that also includes the property of multiplicative inverses, meaning that every element has a unique inverse under multiplication.

2. What are the nine elements of F9?

The nine elements of F9, also known as the Galois field GF(9), are 0, 1, 2, 3, 4, 5, 6, 7, and 8. These elements represent the remainders when dividing by 9, and follow the rules of addition and multiplication in a Galois field.

3. How do you add elements in F9?

In F9, addition is performed by simply adding the elements together and taking the remainder when divided by 9. For example, 3 + 5 = 8 in F9, since 3 + 5 = 8 and 8 divided by 9 leaves a remainder of 8.

4. How do you multiply elements in F9?

Multiplication in F9 is performed by multiplying the elements and taking the remainder when divided by 9. For example, 3 x 5 = 15 in F9, but since 15 divided by 9 leaves a remainder of 6, the product of 3 x 5 in F9 is 6.

5. What is the multiplicative inverse of an element in F9?

The multiplicative inverse of an element in F9 is the number that, when multiplied by the original element, results in a product of 1. For example, the multiplicative inverse of 2 in F9 is 5, since 2 x 5 = 10, which leaves a remainder of 1 when divided by 9.

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