cmi-entrance 2021 Q10

cmi-entrance · India · pgmath 4 marks Groups Ring and Field Structure

Let $K$ be a field of order 243 and let $F$ be a subfield of $K$ of order 3. Pick the correct statement(s) from below.
(A) There exists $\alpha \in K$ such that $K = F ( \alpha )$.
(B) The polynomial $x ^ { 242 } = 1$ has exactly 242 solutions in $K$.
(C) The polynomial $x ^ { 26 } = 1$ has exactly 26 roots in $K$.
(D) Let $f ( x ) \in F [ x ]$ be an irreducible polynomial of degree 5. Then $f ( x )$ has a root in $K$.

Let $K$ be a field of order 243 and let $F$ be a subfield of $K$ of order 3. Pick the correct statement(s) from below.\\
(A) There exists $\alpha \in K$ such that $K = F ( \alpha )$.\\
(B) The polynomial $x ^ { 242 } = 1$ has exactly 242 solutions in $K$.\\
(C) The polynomial $x ^ { 26 } = 1$ has exactly 26 roots in $K$.\\
(D) Let $f ( x ) \in F [ x ]$ be an irreducible polynomial of degree 5. Then $f ( x )$ has a root in $K$.

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