Tag: equazioni
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A simple Diophantine equation
Let’s solve a Diophantine equation together using only elementary considerations. Suppose we are asked to determine all the integers X and Y such that X³-X+1=3Y⁵. As is usually done, we assume that such a solution exists, let’s say (X,Y), and we try to understand if we can make the value of X and Y explicit…
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Equations in algebraically closed fields.
Let K be an algebraically closed field and f∈K[X,Y] non-constant. Let us prove that f has infinitely many zeros in K x K. Let us define the following sets: A={a ∈K : f(a,y) is constant in K[y]}B={b∈K : f(x,b) is constant in K[x] } if one of them is finite we are done. In fact…
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Sum of squares in finite fields
Let K be a finite field. Let us prove that every element of K is the sum of two squares. Let us distinguish two cases. K has characteristic 2. Let us consider the map Φ:K—>K, x |—>x² . Let us observe that Φ is an injective homomorphism. In fact it is a homomorphism having the…
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Fermat Last theorem on ℚ(√2)
Does Fermat’s Last Theorem hold in ℚ(√2)? No, in fact it can be shown that if n>3 then the equation Xⁿ+Yⁿ=Zⁿ has no non-trivial solutions in ℚ(√2,√3) (therefore not even in ℚ(√2) however for n=3 we have that: (18+17√2)³+(18-17√2)³=42³ Source: https://eprints.whiterose.ac.uk/194494/14/Fermat.pdf, https://math.stackexchange.com/questions/4438180/does-fermats-last-theorem-imply-sqrt2-not-in-mathbbq
Marco Damele 