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"text": "The fundamental theorem of algebra states that the field of complex numbers, #x5C;mathbb#x7B;C#x7D;, is algebraically closed.",
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"ref": "2007, Pierre Antoine Grillet, Abstract Algebra, 2nd edition, Springer, page 166",
"text": "Definition. A field is algebraically closed when it satisfies the equivalent conditions in Proposition 4.1. #x5C;square\nFor instance, the fundamental theorem of algebra (Theorem III.8.11) states that #x5C;Complex is algebraically closed. The fields #x5C;R, #x5C;Q, #x5C;Z#x5F;p, are not algebraically closed, but #x5C;R and #x5C;Q can be embedded into the algebraically closed field #x5C;Complex.",
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"text": "In many ways #x5C;Q#x5F;p is analogous to #x5C;R. For example, #x5C;R is not algebraically closed. The exercises below show that #x5C;Q#x5F;p is not algebraically closed. However, by adjoining i#x3D;#x5C;sqrt#x7B;-1#x7D; to #x5C;R, we get the field of complex numbers, which is algebraically closed. In contrast, the algebraic closure #x5C;overline#x5C;Q#x5F;p of #x5C;Q#x5F;p is not of finite degree over #x5C;Q. Moreover, #x5C;Complex is complete with respect to the extension of the usual norm of #x5C;R. Unfortunately, #x5C;overline#x5C;Q#x5F;p is not complete with respect to the extension of the p-adic norm. So after completing it (via the usual method of Cauchy sequences) we get a still larger field, usually denoted by #x5C;Complex#x5F;p, and it turns out to be both algebraically closed and complete.",
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"text": "Definition 5.3.14. Let F be a field. A field extension P is called an algebraic closure of F if P is algebraically closed and every proper subfield of P containing F is not algebraically closed.\nIn other words, the algebraic closure of F is the minimal algebraically closed field containing F.",
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"sense": "that contains among its elements every root of every nonconstant single-variable polynomial definable over it",
"word": "algébriquement clos"
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"sense": "that contains among its elements every root of every nonconstant single-variable polynomial definable over it",
"word": "algebricamente chiuso"
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"sense": "that contains among its elements every root of every nonconstant single-variable polynomial definable over it",
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"sense": "that contains among its elements every root of every nonconstant single-variable polynomial definable over it",
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"text": "In many ways #x5C;Q#x5F;p is analogous to #x5C;R. For example, #x5C;R is not algebraically closed. The exercises below show that #x5C;Q#x5F;p is not algebraically closed. However, by adjoining i#x3D;#x5C;sqrt#x7B;-1#x7D; to #x5C;R, we get the field of complex numbers, which is algebraically closed. In contrast, the algebraic closure #x5C;overline#x5C;Q#x5F;p of #x5C;Q#x5F;p is not of finite degree over #x5C;Q. Moreover, #x5C;Complex is complete with respect to the extension of the usual norm of #x5C;R. Unfortunately, #x5C;overline#x5C;Q#x5F;p is not complete with respect to the extension of the p-adic norm. So after completing it (via the usual method of Cauchy sequences) we get a still larger field, usually denoted by #x5C;Complex#x5F;p, and it turns out to be both algebraically closed and complete.",
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"sense": "that contains among its elements every root of every nonconstant single-variable polynomial definable over it",
"word": "algébriquement clos"
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"sense": "that contains among its elements every root of every nonconstant single-variable polynomial definable over it",
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"sense": "that contains among its elements every root of every nonconstant single-variable polynomial definable over it",
"word": "algebraiskt lukket"
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"sense": "that contains among its elements every root of every nonconstant single-variable polynomial definable over it",
"word": "algebricamente fechado"
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"sense": "that contains among its elements every root of every nonconstant single-variable polynomial definable over it",
"word": "algebraicamente cerrado"
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