Tag: Algebra

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Let’s solve the Diophantine equation X⁴-Y⁴-X=0 together so as to see a standard approach that is encountered in solving Diophantine equations of the form Z(f(Z)-1)=Y² since f is a function of Z. For any typos I invite you to report them in the comments. Let’s suppose that (X,Y) is a non-trivial solution, that is, X…

A simple Diophantine equation

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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…

Automorfismi di ℚ(√2)

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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:…

σ-complete groups

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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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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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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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