Uploaded September 2020 | Updated September 2026, 1 week ago
Proof that quadratic residues always have even index: youtu.be/zs0ZtQzVBSE
Intro to indices: youtu.be/jLeNX2jYuUs
One of the most important properties of the Legendre symbol is that we can split up the Legendre symbol of a product. Using the power of indices, we can easily prove how this works!
Quadratic Residues playlist: youtube.com/playlist?list=PLug5ZIRrShJGJieICD2KnqU9cRioaEBjB
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Music: OcularNebula - The Lopez
Proof that quadratic residues always have even index: youtu.be/zs0ZtQzVBSE
Intro to indices: youtu.be/jLeNX2jYuUs
One of the most important properties of the Legendre symbol is that we can split up the Legendre symbol of a product. Using the power of indices, we can easily prove how this works!
Quadratic Residues playlist: youtube.com/playlist?list=PLug5ZIRrShJGJieICD2KnqU9cRioaEBjB
Subscribe to see more new math videos!
Music: OcularNebula - The Lopez
![How the Matrix Characteristic Polynomial is Connected to F[x]-Modules](https://i.ytimg.com/vi/jCt6mR3QtPk/mqdefault.jpg "How the Matrix Characteristic Polynomial is Connected to F[x]-Modules
Rational canonical form: https://youtu.be/q5uj4o0O5R0
Similar matrices isomorphism: https://youtu.be/-ligAAxFM8Y
Intro to F[x]-modules: https://youtu.be/H44q_Urmts0
The characteristic polynomial described by the determinant det(xI-A) is very useful when studying matrices. It turns out that its related to modules as well! Here we show how the characteristic polynomial is related to the module form of an arbitrary matrix.
Ring & Module Theory playlist: https://www.youtube.com/playlist?list=PLug5ZIRrShJExMapwnaKTFXDYbKeWDXq7
0:00 Computing the determinant
10:07 Similar matrices have same polynomial
12:40 Proof for a general matrix
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Music: C418 - Pr Department")
