Newbetuts

Prove that the fields $\mathbb Z_{11}[x]/\langle x^2+1\rangle$ and $\mathbb Z_{11}[x]/\langle x^2+x+4 \rangle$ are isomorphic

Let I and J be ideals of a ring R. Show by example that the set of products {xy | x ∈ I, y ∈ J} need not be an ideal

Prove that the set $a+b\sqrt{2}$ where a and b are rational without zero is a group under multiplication [duplicate]

Can we axiomatize a field starting with the binary operations and only “equational” axioms?

Finite fields are isomorphic

Polynomials having as roots the sum (respectively, the product) of two algebraic elements

Showing that $\sqrt[3]{2}\notin\Bbb Q(\alpha_1,...,\alpha_k)$ where $\alpha_i^2\in\Bbb Q\ \forall i$

Can a field be isomorphic to its subfield?

If $E$ is generated by $E_1$ and $E_2$, then $[E:F]\leq [E_1:F][E_2:F]$?

Is it true that $\mathbb{C}(x) \equiv \mathbb{C}(x, y)$?

Is $\mathbb{R}[X]/(P)$ isomorphic to $\mathbb C$ for every irreducible polynomial $P$ of degree $2?$

What is the overall idea of Galois theory?

Are logarithms of prime numbers quadratically independent over $\mathbb Q$?

Is $\bar{\mathbb{Q}}(x)\cap \mathbb{Q}((x))=\mathbb{Q}(x)$? [unsolved (even though we earlier thought it was)]

Multiplicative inverse of a quadratic algebraic number $\,a+b\sqrt 2$

New posts in field-theory

Does there exist a field $(F,+,)$ so that $(F,+) \cong (F^,*)$?

Category of Field has no initial object

Minimal Polynomial of $\sqrt{2}+\sqrt{3}+\sqrt{5}$

$p^{th}$ roots of a field with characteristic $p$

Algebraic extensions and sub rings

Prove that the fields $\mathbb Z_{11}[x]/\langle x^2+1\rangle$ and $\mathbb Z_{11}[x]/\langle x^2+x+4 \rangle$ are isomorphic

Let I and J be ideals of a ring R. Show by example that the set of products {xy | x ∈ I, y ∈ J} need not be an ideal

Prove that the set $a+b\sqrt{2}$ where a and b are rational without zero is a group under multiplication [duplicate]

Can we axiomatize a field starting with the binary operations and only “equational” axioms?

Finite fields are isomorphic

Polynomials having as roots the sum (respectively, the product) of two algebraic elements

Showing that $\sqrt[3]{2}\notin\Bbb Q(\alpha_1,...,\alpha_k)$ where $\alpha_i^2\in\Bbb Q\ \forall i$

Can a field be isomorphic to its subfield?

If $E$ is generated by $E_1$ and $E_2$, then $[E:F]\leq [E_1:F][E_2:F]$?

Is it true that $\mathbb{C}(x) \equiv \mathbb{C}(x, y)$?

Is $\mathbb{R}[X]/(P)$ isomorphic to $\mathbb C$ for every irreducible polynomial $P$ of degree $2?$

What is the overall idea of Galois theory?

Are logarithms of prime numbers quadratically independent over $\mathbb Q$?

Is $\bar{\mathbb{Q}}(x)\cap \mathbb{Q}((x))=\mathbb{Q}(x)$? [unsolved (even though we earlier thought it was)]

Multiplicative inverse of a quadratic algebraic number $\,a+b\sqrt 2$