Marco Damele

Category: Un pò di matematica

  • Automorfismi di ℚ(√2)

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    Marco Damele
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    Ricordiamo che ℚ(√2) = {x+y√2, x,y∈ℚ} e questo è un campo con le operazioni: (a+b√2)+(a’+b’√2)=(a+a’)+(b+b’)√2(a+b√2)(a’+b’√2)=(aa’+2bb’)+(ab’+ba’)√2 Un automorfismo di ℚ(√2) è una mappaφ: ℚ(√2) —> ℚ(√2) bigettiva tale che ∀ A,B∈ ℚ(√2) risulta: φ(A+B)=φ(A)+φ(B)φ(AB)=φ(A)φ(B) TEOREMA Tutti gli automorfismi del campo ℚ(√2) sono dati da:F:ℚ(√2) —> ℚ(√2) , x+y√2|—> x+y√2 eG:ℚ(√2) —> ℚ(√2) , x+y√2|—> x-y√2 DIMOSTRAZIONE:…

  • Somme di radici di numeri razionali.

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    Marco Damele
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    Dimostriamo che se n ∈ ℕ con n>1 la somma 1+√2+…+√n è irrazionale. Per farlo proviamo il seguente fatto più generale: Dati m1,…,mk in R⁺ tali che √m1 + … + √mk ∈ ℚ —> √mj ∈ ℚ per ogni j=1,…,k. Questo proverebbe che se la somma 1+√2+…+√n ∈ ℚ allora , ad esempio, √2…

  • Horned sphere

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    Marco Damele
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    Let A,B ⊆ IRⁿ be closed and homeomorphic as topological spaces equipped with the topology induced by the Euclidean one of IRⁿ. It is known that any homeomorphism h:A—>B extends to a homomorphism φ:IR²ⁿ—>IR²ⁿ. It follows that IR²ⁿ \ A is homeomorphic to IR²ⁿ \ φ(A) = IR²ⁿ \ h(A) = IR²ⁿ \ B .…

  • σ-complete groups

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    Marco Damele
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    Consider a finite group G. In general it is interesting to know when, given that N is a maximal normal subgroup of G, we have G ≅ N x G/N. Clearly this does not always happen. For example, take a prime p and G=Zp². The maximal proper normal subgroup is Zp and G/N=Zp but clearly…

  • Equations in algebraically closed fields.

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    Marco Damele
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    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…

  • Sum of squares in finite fields

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    Marco Damele
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    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…

  • Fermat Last theorem on ℚ(√2)

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    Marco Damele
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    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