Recent questions tagged field-theory

Let $K$ be the splitting field of $X^{n}-1$ over $\mathbb{F}_{p}$, where $n$ is a positive integer. Pick the correct statement(s) from below.$K$ has $p^{n}$ elements.If $...

If $f: F \rightarrow F$ is a homomorphism of fields, then $f$ is surjective.

Let $X$ be a non-empty finite set and let $R$ be the ring of $\mathbb{Z}$-valued functions on $X$, with pointwise addition and multiplication. Let $S$ be an additive subg...

Let $p \geq 3$ be a prime number and $V$ be an $n$-dimensional vector space over $\mathbb{F}_{p}$. Let $T: V \rightarrow V$ be a linear transformation. Select all the tru...

Which of the following statement(s) are true?If $F_{1}, F_{2}$ are finite field extensions of $\mathbb{Q}$ such that $\left[F_{1}: \mathbb{Q}\right]=\left[F_{2}: \mathbb{...

Let $p, q$ be distinct prime numbers and let $\zeta_{p}, \zeta_{q}$ denote (any) primitive $p$-th and $q$-th roots of unity, respectively. Choose all the correct statemen...

What is the number of distinct subfields of $\mathbb{C}$ isomorphic to $\mathbb{Q}[\sqrt[3]{2}]$?$1$$2$$3$Infinite