Note: \"a>n-1\" means \"a subscript (n-1)\" A field F is called algebraically cl

Question

Note: "a>n-1" means "a subscript (n-1)"

A field F is called algebraically closed if every algebraic equation (x^n) + (a>n-1)(x^n-1)+...+(a>1)(x) + a>0 = 0 where (a>n-1), (a>n-2),...,a>1, a>0 is an element of F and n is greater than or equal to 1 has a root in F. For instance, Fundamental Theorem of Algebra implies thatthe set of allComplex numbers is algebraically closed. But, the set of all Real numbers is not algebraically closed (what are the roots of x^2 + 1 =0 ?). *Show that a finite field F is not algebraically closed.*

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Explanation / Answer

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F[x], the ring of polynomials in the variable x with coefficients in F. Examples As an example, the field of real numbers is not algebraically closed, because the polynomial equation x2 + 1 = 0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. Also, no finite field F is algebraically closed, because if a1, a2,

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