What are the roots of x^(p-1) in Z_p?

In summary, the process for finding the root of a polynomial involves setting the polynomial equal to zero and using methods such as factoring, graphing, or using the quadratic formula. A polynomial can have real or complex roots, depending on the solutions to the polynomial equation. It is possible for a polynomial to have more than one root, with the number of roots being equal to its degree. Finding the root of a polynomial is significant because it allows us to solve for the values of the variable and make predictions about its behavior. There are special cases when finding the root of a polynomial, such as polynomials with a degree of zero or one, and those with repeated or imaginary roots.
  • #1
kathrynag
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Let p be a prime number. Find all roots of x^(p-1) in Z_p



I have this definition.
Let f(x) be in F[x]. An element c in F is said to be a root of multiplicity m>=1 of f(x) if (x-c)^m|f(x), but (x-c)^(m+1) does not divide f(x).

I'm not sure if I use this idea somehow or not.
 
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  • #2
I think Fermat's Little Theorem will be useful here.
 

What is the process for finding the root of a polynomial?

The process for finding the root of a polynomial involves setting the polynomial equal to zero and then using different methods such as factoring, graphing, or using the quadratic formula to solve for the value(s) of the variable that make the polynomial equal to zero.

How do you know if a polynomial has real or complex roots?

A polynomial will have real roots if the solutions to the polynomial equation are real numbers. A polynomial will have complex roots if the solutions to the polynomial equation involve imaginary numbers, such as the square root of a negative number.

Can a polynomial have more than one root?

Yes, a polynomial can have multiple roots. The number of roots a polynomial has is equal to its degree, or the highest exponent of the variable in the polynomial. For example, a quadratic polynomial (degree 2) can have up to two roots.

What is the significance of finding the root of a polynomial?

Finding the root of a polynomial is important because it allows us to solve for the values of the variable that make the polynomial equation true. This can help us understand the behavior of the polynomial and make predictions about its graph.

Are there any special cases when finding the root of a polynomial?

Yes, there are a few special cases when finding the root of a polynomial. For example, a polynomial with a degree of zero has no root, and a polynomial with a degree of one has only one root. Additionally, some polynomials may have repeated roots or imaginary roots.

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