Find the splitting field x4 - x2 - 2 over Z3.
Find the splitting field x4 - x2 - 2 over Z3.
In mathematics, a splitting field is a field extension that contains all of the roots of a given polynomial. In this problem, we are asked to find the splitting field of the polynomial :
x⁴ - x² - 2
over the field Z3 . To solve this problem, we will need to use some basic concepts from field theory and algebraic extensions, such as primitive elements, roots of polynomials, and bases for fields. By following a series of steps, we will be able to determine the splitting field of the given polynomial and express it in terms of a suitable basis.
Rewrite the polynomial x⁴ - x² - 2 in terms of a primitive element:
The first step is to rewrite the polynomial x⁴ - x² - 2 in terms of a primitive element of the field Z₃. A primitive element is an element that generates the entire field through repeated multiplication.
In this case, we can choose the primitive element to be x. This means that we can rewrite the polynomial as :
x⁴ - x² - 2 = 0
The next step is to find the roots of the polynomial. These roots will form the basis for the splitting field.
To find the roots, we can try substituting different values for x into the polynomial until we find a value that makes the polynomial equal to 0.
For example, if we substitute x = 0 into the polynomial, we get:
0⁴ - 0² - 2 = -2
Since -2 is not equal to 0, x = 0 is not a root of the polynomial.
If we substitute x = 1 into the polynomial, we get:
14 - 12 - 2 = -1
Since -1 is not equal to 0, x = 1 is not a root of the polynomial.
If we substitute x = 2 into the polynomial, we get:
24 - 22 - 2 = 2
Since 2 is not equal to 0, x = 2 is not a root of the polynomial.
At this point, we can conclude that there are no roots of the polynomial in the field Z₃.
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