2015-12-4: Difference between revisions

Revision as of 16:27, 2 September 2023

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Problema:} Fie $K$ un corp cu $m\geq 2$ elemente si Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \in K[X]} . Aratati ca urmatoarele afirmatii sunt echivalente:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (i)} Exista Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in K[X]} astfel incat Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(X)=g(X^{m-1})} ;

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (ii)} Pentru orice Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in K^*} avem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(X)=f(aX)} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Solutie}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (i) \rightarrow (ii)} Din teorema lui Lagrange aplicata grupului Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (K^*,\cdot)} avem ca Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{m-1}=1, \forall x \in K^*} , deci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(aX)=g((aX)^{m-1})=g(a^{m-1}X^{m-1})=g(X^{m-1})=f(X)} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (ii) \rightarrow (i)}

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 <math>Problema:</math> Fie <math>K</math> un corp cu <math>m \geq 2</math> elemente si <math>f \in K[X]</math>. Aratati ca urmatoarele afirmatii sunt echivalente:
-(i) Exista <math>g \in K[X]</math> astfel incat <math>f(X)=g(X^{m-1})</math>;
+<math>(i)</math> Exista <math>g \in K[X]</math> astfel incat <math>f(X)=g(X^{m-1})</math>;
-(ii) Pentru orice <math>a \in K^*</math> avem <math>f(X)=f(aX)</math>.
+<math>(ii)</math> Pentru orice <math>a \in K^*</math> avem <math>f(X)=f(aX)</math>.
 <math>Solutie</math>
 <math>(i) \rightarrow (ii)</math>
-Din teorema lui Lagrange aplicata grupului <math>(K^*,\cdot)</math> avem ca <math>x^{m-1}=1, \forall x \in K^*</math>, deci <math>f(aX)=g((aX)^{m-1}=g(a^{m-1}X^{m-1})=g(X^{m-1})=f(X)</math>.
+Din teorema lui Lagrange aplicata grupului <math>(K^*,\cdot)</math> avem ca <math>x^{m-1}=1, \forall x \in K^*</math>, deci <math>f(aX)=g((aX)^{m-1})=g(a^{m-1}X^{m-1})=g(X^{m-1})=f(X)</math>.
+<math>(ii) \rightarrow (i)</math>