Answer to Question #202566 in Abstract Algebra for Maheen Fatima

Question

Write ten examples of Extension field?

Answer 1

$(1)The\space field\space of\space complex\space numbers\space {\displaystyle\space \mathbb\space }\mathbb{C}\space \space is\space an\space extension\space field\space of\space the\space field\space of\\\space real\space numbers\space {\displaystyle \mathbb\space }\mathbb\space {R}\space$

$(2)The\space field\space of\space complex\space numbers\space {\displaystyle\space \mathbb\space }\mathbb{C}\space \space is\space an\space extension\space field\space of\space the\space field\space of\\\space rational\space numbers\space {\displaystyle \mathbb\space }\mathbb\space {Q}\space$

$(3)The\space field\space of\space real\space numbers\space {\displaystyle\space \mathbb\space }\mathbb{R}\space \space is\space an\space extension\space field\space of\space the\space field\space of\\\space rarional\space numbers\space {\displaystyle \mathbb\space }\mathbb\space {Q}\space$

$(4)The\space field\space {\displaystyle \mathbb {Q} ({\sqrt {2}})=\left\{a+b{\sqrt {2}}\mid a,b\in \mathbb {Q} \right\},} is \space an \space extension \space field\space of\space {\displaystyle \mathbb {Q} }$

$(5)The \space field \\ {\displaystyle {\begin{aligned}\mathbb {Q} \left({\sqrt {2}},{\sqrt {3}}\right)&=\mathbb {Q} \left({\sqrt {2}}\right)\left({\sqrt {3}}\right)\\&=\left\{a+b{\sqrt {3}}\mid a,b\in \mathbb {Q} \left({\sqrt {2}}\right)\right\}\\&=\left\{a+b{\sqrt {2}}+c{\sqrt {3}}+d{\sqrt {6}}\mid a,b,c,d\in \mathbb {Q} \right\},\end{aligned}}}\\ is\space an \space extension\space field \space of\space {\displaystyle \mathbb {Q\sqrt{2}} }$

$(6)The \space field \\ {\displaystyle {\begin{aligned}\mathbb {Q} \left({\sqrt {2}},{\sqrt {3}}\right)&=\mathbb {Q} \left({\sqrt {2}}\right)\left({\sqrt {3}}\right)\\&=\left\{a+b{\sqrt {3}}\mid a,b\in \mathbb {Q} \left({\sqrt {2}}\right)\right\}\\&=\left\{a+b{\sqrt {2}}+c{\sqrt {3}}+d{\sqrt {6}}\mid a,b,c,d\in \mathbb {Q} \right\},\end{aligned}}}\\ is\space an \space extension\space field \space of\space {\displaystyle \mathbb {Q\sqrt{3}} }$

$(7)The \space field \\ {\displaystyle {\begin{aligned}\mathbb {Q} \left({\sqrt {2}},{\sqrt {3}}\right)&=\mathbb {Q} \left({\sqrt {2}}\right)\left({\sqrt {3}}\right)\\&=\left\{a+b{\sqrt {3}}\mid a,b\in \mathbb {Q} \left({\sqrt {2}}\right)\right\}\\&=\left\{a+b{\sqrt {2}}+c{\sqrt {3}}+d{\sqrt {6}}\mid a,b,c,d\in \mathbb {Q} \right\},\end{aligned}}}\\ is\space an \space extension\space field \space of\space {\displaystyle \mathbb {Q} }$

$(8)\\{\displaystyle {\begin{aligned}\mathbb {} { {}}{ {}}&\mathbb {Q} ({\sqrt {2}}+{\sqrt {3}})\\&=\left\{a+b({\sqrt {2}}+{\sqrt {3}})+c({\sqrt {2}}+{\sqrt {3}})^{2}+d({\sqrt {2}}+{\sqrt {3}})^{3}\mid a,b,c,d\in \mathbb {Q} \right\}.\end{aligned}}}\\ is\space an \space extension\space field \space of\space {\displaystyle \mathbb {Q} }$

$(9) F\space is\space any\space field\space and\space F[x]\space the\space polynomial\space ring.\space \\Let\space F(X)\space be\space the quotient\space field\space of\space F[x].\space Then,\space F(X)\space is\space an\space extension\space field\space of\space F.$

$(10)any\space field\space F,\space F\space is\space a\space finite\space extension\space over\space itself.$