Tag: campi
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Let G be a group such that Aut(G) is cyclic and not trivial. Then Aut(G) is finite and his order is even.
Since Aut(G) is cyclic then Inn(G) is cyclic so G/Z(G) is cyclic which implies G being abelian. Let us first show that Aut(G) has an element of order 2. Consider the map: f: x ∈ G |—> x⁻¹ ∈ G we have (G being abelian) f(xy)=(xy)⁻¹=y⁻¹x⁻¹=x⁻¹y⁻¹=f(x)f(y) so f is a group morphism. Now f is…
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Let f ∈ ℚ[X] irreducible, then f doesn’t have multiple roots in ℂ
To prove the assertion suppose f=a₀+a₁X+…+anXⁿ and let α be a multiple root of f. Consider g=an⁻¹f . Then g is irreducible, monic and has α has a multiple root (which make, by the way, g the minimal polynomial over ℚ of α) . Then α has to be a root of g’ (the derivative…
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Six proofs that √2 is irrational
Dimostriamo che √2 è irrazionale usando 6 dimostrazioni diverse. 1° Proof: Suppose that √2 ∈ ℚ then there exist two coprime integers a and b such that √2=a/b. It follows that 2b²=a². So a² is even, that is, a is even, or a=2k with k being an integer. But then 2b²=4k², or b²=2k², from which…
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If K is a field (K,+) it is not isomorphic to (K{0},•).
Let (K,+,•) be a field, let us prove that its additive group (K,+) is not isomorphic to the group of invertibles of K, that is (K{0},•). Let us suppose that there is an isomorphism between them, then, since isomorphisms preserve the order of an element, we would have that: |{x ∈ K : x+x=0}|=|{x∈ K{0}…
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Ring with p elements
Let us prove that every (unitary) ring R with p elements, p being prime, is isomorphic to ℤp. Let us consider the homomorphism φ:ℤ—>R , 1|—>1. We have that Ker(φ) is an ideal of ℤ so Ker(φ)=nℤ for some n∈ℕ. Now by the first isomorphism theorem ℤ/Ker(φ) is isomorphic to Im(φ). By Lagrange’s theorem we…
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Finite groups of class 2 e 3.
Let G be a finite group. We want to classify all finite groups with exactly two and three conjugacy classes. The number of conjugacy classes of G is called the class of G and is denoted by Cl(G). In the following if x ∈ G the conjugacy class of x is denoted by Cl(x). Then…
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Automorfismi di ℚ(√2)
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:…
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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…
Marco Damele 