cmi-entrance 2022 Q4

cmi-entrance · India · pgmath 4 marks Groups True/False with Justification
Pick the correct statement(s) from below.
(A) There exists a finite commutative ring $R$ of cardinality 100 such that $r ^ { 2 } = r$ for all $r \in R$.
(B) There is a field $K$ such that the additive group ( $K , +$ ) is isomorphic to the multiplicative group ( $K ^ { \times } , \cdot$ ).
(C) An irreducible polynomial in $\mathbb { Q } [ x ]$ is irreducible in $\mathbb { Z } [ x ]$.
(D) A monic polynomial of degree $n$ over a commutative ring $R$ has at most $n$ roots in $R$.
Pick the correct statement(s) from below.\\
(A) There exists a finite commutative ring $R$ of cardinality 100 such that $r ^ { 2 } = r$ for all $r \in R$.\\
(B) There is a field $K$ such that the additive group ( $K , +$ ) is isomorphic to the multiplicative group ( $K ^ { \times } , \cdot$ ).\\
(C) An irreducible polynomial in $\mathbb { Q } [ x ]$ is irreducible in $\mathbb { Z } [ x ]$.\\
(D) A monic polynomial of degree $n$ over a commutative ring $R$ has at most $n$ roots in $R$.