When you drive on a bumpy road, your CD music plays normally and smoothly while your body bumps up and down in your car?

Why CD disk can be bumpy-free ?

Thanks to Galois, the 19-year-old French Math genius and tragic victim of the X Concours. Galois invented the whole Abstract Algebra with two revolutionary ideas: Group and Field.

Galois Group explains why symmetry is a Math structure which plays the role of Quintic equation with no radical solutions.

**Galois Field**is used in the cryptography coding for computers, digital music in CD / DVD.

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**Galois Field Theorem:**

*the only one in Field Theory*

There is EXACTLY ONE such system whenever the number of element is a power of any prime number p.

<=>

Fp contains EXACTLY 1 sub-Field with pⁿ elements. (n ≥1)

<=>

For p prime, n ≥1

|Fp | = pⁿ unique

Example:

p=2 n =1

=> Galois Field = {0,1} with pⁿ = 2 elements

** Galois Field {0,1}**

1+1 = 0 = 0+0

1+0 = 1 = 0 + 1

**Reed-Solomon codes for CD, DVD**

(1960) using binary Galois Field {0,1}

Error correcting codes based on algebraic properties of Polynomials, whose coefficients are taken from a Galois Field.

255 bytes block = 223 + 32

223 = encode the signal

32 bytes = parity codes

When the car bumps and so the CD player’s laser beam on the CD disk, digital music code errors can be auto-corrected using

*Reed-Solomon code Algorithm, so you don’t hear any jerking disruption.*
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