[Number theory] Calculate the Hilbert symbol

[nonequilibrium]

Homework Statement

Determine the Hilbert symbol $\left( \frac{2,0}{\mathbb F_{25}} \right)$ where the F denotes the field with 5² elements.

Homework Equations

$\left( \frac{2,0}{\mathbb F_{5}} \right) = -1$

The Attempt at a Solution

Due to the formula that I put under "relevant equations", we know that the polynomial f(x) = 2x²-1 has NO solution in F_5, hence it is irreducible. Look at the splitting field of this polynomial, call it X. Then by construction $\left( \frac{2,0}{X} \right) = 1$. Now note that since the degree of f is 2, X is a vector space over F_5 of dimension 2 and hence has to be isomorphic to F_25. This concludes the proof.

(The last part is what I'm unsure about; does it require more argumentation? And is there perhaps an even shorter way of showing that the Hilbert symbol is 1?)

 

[mighty2000]

Thank you for your question. The Hilbert symbol is a mathematical tool used to determine the solvability of quadratic equations over finite fields. In this case, we are looking at the field with 5² elements, denoted as F_25. The Hilbert symbol is defined as follows:

\left( \frac{a,b}{\mathbb F_{p^n}} \right) = \begin{cases} 1 & \text{if } ax^2+by^2 = 1 \text{ has a solution in } \mathbb F_{p^n} \\ -1 & \text{if } ax^2+by^2 = 1 \text{ has no solution in } \mathbb F_{p^n} \end{cases}

In this case, we are looking at the Hilbert symbol \left( \frac{2,0}{\mathbb F_{25}} \right). Using the relevant equation provided, we can see that \left( \frac{2,0}{\mathbb F_{5}} \right) = -1. This means that the polynomial f(x) = 2x²-1 has no solution in F_5, making it irreducible.

To determine the Hilbert symbol for \left( \frac{2,0}{\mathbb F_{25}} \right), we need to look at the splitting field of f(x) over F_5, denoted as X. Since X is a vector space over F_5 of dimension 2, it must be isomorphic to F_25. Therefore, \left( \frac{2,0}{X} \right) = 1.

This means that the polynomial f(x) = 2x²-1 has a solution in F_25, making the Hilbert symbol \left( \frac{2,0}{\mathbb F_{25}} \right) = 1. This concludes the proof.

I hope this helps to clarify your doubts. Let me know if you have any further questions. Keep up the good work!