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Field (mathematics): Quiz

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Question 1: The Galois group in this case is obtained as a "limit" (specifically an ________) of the Galois groups of the finite Galois extensions of E.
Inverse limitDirect limitGroup (mathematics)Category (mathematics)

Question 2: ________integral domainsintegrally closed domainsunique factorization domainsprincipal ideal domainsEuclidean domainsfieldsfinite fields.
Commutative ringField (mathematics)Ring (mathematics)Vector space

Question 3: The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various ________, p-adic fields, and so forth.
Algebraic number theoryField (mathematics)Algebraic number fieldVector space

Question 4: Generalizing in a more categorical direction yields the ________ and related objects.
Field with one elementVector spaceAlgebraic geometryBuilding (mathematics)

Question 5: The set of all surreal numbers with birthday smaller than some ________ form a field.
Inaccessible cardinalZermelo–Fraenkel set theorySet theoryConstructible universe

Question 6: The concept of a field is of use, for example, in defining vectors and matrices, two structures in ________ whose components can be elements of an arbitrary field.
Euclidean vectorEigenvalue, eigenvector and eigenspaceLinear algebraDual space

Question 7: The concept of field was used implicitly by ________ and Évariste Galois in their work on the solvability of polynomial equations with rational coefficients of degree 5 or higher.
Christopher HansteenCarl Friedrich GaussAdrien-Marie LegendreNiels Henrik Abel

Question 8: Assuming the ________, for every field F, there exists a field F, called the algebraic closure of F, which contains F, is algebraic over F, which means that any element x of F satisfies a polynomial equation
Set theoryAxiom of choiceMathematical logicZermelo–Fraenkel set theory

Question 9: From the point of view of ________, fields are points, because the spectrum Spec F has only one point, corresponding to the 0-ideal.
CalculusMathematicsAlgebraic geometryNumber theory

Question 10: If X is an ________ over F, then the rational functions VF, i.e., functions defined almost everywhere, form a field, the function field of V.
Scheme (mathematics)Projective spaceAlgebraic geometryAlgebraic variety







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