Grupurile (K,+) si (K, .) nu pot fi izomorfe
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Grupurile (K,+) si (K, .) nu pot fi izomorfe
Aratati ca daca \( (K, +, \cdot) \) este un corp, atunci grupurile \( (K, +) \) si \( (K*, \cdot ) \) nu pot fi izomorfe.
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second orders half a beer. The third, a quarter of a beer. The bartender says “You’re all idiots”, and pours two beers.
Presupunem ca \( (K,+)\approx (K^*,\cdot) \)
Ar fi doua cazuri
1. \( char K \neq 2 \)
Consideram ecuatiile \( x+x=0 \) in \( (K,+) \) (1) si \( x\cdot x=1 \) in \( (K^*,\cdot) \) (2). Ecuatia (1) are o solutie \( x=0 \), iar ecuatia (2) are doua solutii \( x\in\left{-1,1\right} \) (\( 1\neq -1 \)). (absurd!)
2. \( char K=2 \)
\( x+x=0 \Rightarrow x^2=1, \forall x\in K^* \). \( (x+1)^2=x^2+1=1+1=0 \Rightarrow x^2=0, \forall x\in K \) .(absurd!)
Ar fi doua cazuri
1. \( char K \neq 2 \)
Consideram ecuatiile \( x+x=0 \) in \( (K,+) \) (1) si \( x\cdot x=1 \) in \( (K^*,\cdot) \) (2). Ecuatia (1) are o solutie \( x=0 \), iar ecuatia (2) are doua solutii \( x\in\left{-1,1\right} \) (\( 1\neq -1 \)). (absurd!)
2. \( char K=2 \)
\( x+x=0 \Rightarrow x^2=1, \forall x\in K^* \). \( (x+1)^2=x^2+1=1+1=0 \Rightarrow x^2=0, \forall x\in K \) .(absurd!)