Tag: anelli
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The Diophantine equation x²-y³=7.
We will prove there are not integer solution to this equation. Suppose that (x,y) is a possible solution of the Diophantine equation. If y is even, reducing modulo 8 we would have x²=7 mod 8, which is absurd because 7 does not fit among the squares modulo 8. It follows that y is odd. Then…
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If m is odd and n a natural number (2ᵐ-1,2ⁿ+1)=1.
In the following i denote with (a,b) the greatest common divisor between two integers a and b. Let p now be a possible prime divisor of A=2ᵐ-1 and B=2ⁿ+1. Let [2]_p denote the class of 2 modulo p. Let us note that since (2,p)=1 then [2]_p is an element of the group of invertibles of…
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There is no group G such that G’=D2n, for n≥3.
Let G be a group. We denote with G’ the commutator subgroup of G and with D2n the diedral group of order 2n, with n a natural number ≥3. Let us prove that there is no group G such that G’=D2n. Suppose that such a group exists, then since G’=<r,s> we get: <r> char G’…
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If R is a local ring then R[[x]] is local.
Recall that if S is a commutative ring with unit, S is local if and only the set of non unit of S is an ideal, i.e, denoting with S* the unit of S, S-S* is an ideal of S. Now, let m be the maximal ideal of R. We prove that R[[x]] – R[[x]]*…
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Let R be a local ring and a,b in R such that (a)=(b). Prove that there exists a unit r of R such that a=rb.
We have a=rb and b=sa for some element r and s of R. Then a=rsa so a(1-rs)=0. Clearly if a=0 then b=0 and we can just take r=1. Suppose a≠0. If r∈R \ R* then, since R is local, r∈J(R) where J is the Jacobson radical of R. Then 1-rs ∈ R* so that a=0,…
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Let F be a field. Can F be realized as a quotient of ℤ[i] ?
Lemma 1: If F ≅ ℤ[i]/J for some ideal J of ℤ[i] then F is a finite field. Proof: It is sufficient to show that ℤ[i]/J is finite. Since ℤ[i] is not a field we can assume J not equal to zero. Since ℤ[i] is a Euclidean domain it is in particular a PID so…
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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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Ring with 4 elements
Let R be a unit ring with 4 elements. Let us prove that R is commutative. R ={0,1,a,b}. Let x,y be in R and let us prove that xy=yx. If x=0 then obviously xy=0=yx. If x=1 then xy=y=yx. We then have to prove that ab=ba. We have a+1∈ R. So a+1=0 or a+1=b. If a+1=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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Can a polynomial of degree n have more than n roots?
Let’s start with some definitions. Let R be a ring and f∈R [X] a polynomial of degree n≥0, then f=a0+a1X+…+anXⁿ where a0,a1,…,an are elements of R. An element α∈R is called a root of f if f(α)=0, that is, if a0+a1α+…+anαⁿ=0. For example, the polynomial X²-2 ∈ ℝ[X] has as its only roots √2 and…
Marco Damele 