Write ten examples of Extension field?
"(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 \n\n{\\displaystyle \\mathbb {Q} ({\\sqrt {2}})=\\left\\{a+b{\\sqrt {2}}\\mid a,b\\in \\mathbb {Q} \\right\\},}\nis \\space an \\space extension \\space field\\space of\\space {\\displaystyle \\mathbb {Q} }"
"(5)The \\space field \\\\\n{\\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}}}\\\\\nis\\space an \\space extension\\space field \\space of\\space {\\displaystyle \\mathbb {Q\\sqrt{2}} }"
"(6)The \\space field \\\\\n{\\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}}}\\\\\nis\\space an \\space extension\\space field \\space of\\space {\\displaystyle \\mathbb {Q\\sqrt{3}} }"
"(7)The \\space field \\\\\n{\\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}}}\\\\\nis\\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}}}\\\\\nis\\space an \\space extension\\space field \\space of\\space {\\displaystyle \\mathbb {Q} }"
"(9)\nF\\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\nquotient\\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."
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