Question

Linear Algebra

Recall that Z denotes the ring of integers, and Q is the rationals. We let F[x]denote the polynomial ring in the variable.in F[x].over a field F, and we let f(x) = x²+1

\text { (1) Show that } f(x) \text { is irreducible when } F=Z /(7) \text {. }

Fx]/(f(x)) with F = Z/(3), and let a e E be a root of f(x).(2) Let E =Express g(1+a) when g(x) = 2x²+x²+1 in the basis {1, a} of the F-vector space E.

\text { (3) Assume that the integer } p \text { is a prime of the form } p=a^{2}+b^{2} \text {. Let } F^{\prime}=Z /(p) \text {. }

\text { Is the complex number } \frac{1}{4}(2+i \sqrt{3}) \text { algebraic (over the rationals)? }

\text { Is } x^{5}+5 x+5 \text { irreducible in } \mathrm{Q}[x] ?

\text { Factor } x^{4}+x^{2}+1 \text { into irreducible factors in } Q[x] \text {. }

') Determine the degree of the extension Q C E when E is the splitting field of the rational polynomial x4 + x² + 1

Show that a finite field F can not be algebraically closed.


Answer

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