List questions from extension-field

Published on 25 Mar 2026 - 4:57

872

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Show that $F(a) = F(a^{2})$ knowing that $F(a)$ has odd degree.

Published on 05 Apr 2018 - 13:36

449

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Decomposing an abelian extension into cyclic extensions

Published on 25 Mar 2026 - 6:00

1k

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Prove that the polynomial $x^4+3x+3$ is irreducible over the field $\mathbb{Q}[2^{1/3}]$

Published on 05 Apr 2018 - 20:02

70

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If $K$ and $K'$ are extensions of the same field $F$, is every isomorphism $K \to K'$ a $F$-isomorphism?

Published on 05 Apr 2018 - 20:02

1.1k

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Algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$. Alternative proof?

Published on 05 Apr 2018 - 20:32

107

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Prove that $[\mathbb{Q}(\sqrt[4]{2}, \sqrt{3}):\mathbb{Q}] = 8$

Published on 05 Apr 2018 - 21:00

105

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Composite of field extensions

Published on 06 Apr 2018 - 0:51

292

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Proving the lemma $\sqrt2\notin\mathbb{Q}(\sqrt3,\sqrt5)$

Published on 06 Apr 2018 - 1:19

160

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Proof that $[\overline{\mathbb Q}:\mathbb R\cap\overline{\mathbb Q}]=2$

Published on 06 Apr 2018 - 8:39

185

Views

Noether's Normalization Lemma states something that appears to contradict the Nullstellensatz. Where have I gone wrong?

Published on 06 Apr 2018 - 11:32

40

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Examples in finite extensions.

Published on 07 Apr 2018 - 19:55

1.1k

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Galois Group of splitting field of $x^4 - x -1$ over $\mathbb{Q}$?

Published on 08 Apr 2018 - 20:59

134

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Proof verification. Question about finite extension, minimal polynomial.

Published on 09 Apr 2018 - 22:38

68

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A doubt in understanding a proof in Galois Theory

Published on 12 Apr 2018 - 12:37

List Question

Quickest way to prove irreduciblity?

Show that $F(a) = F(a^{2})$ knowing that $F(a)$ has odd degree.

Decomposing an abelian extension into cyclic extensions

Prove that the polynomial $x^4+3x+3$ is irreducible over the field $\mathbb{Q}[2^{1/3}]$

If $K$ and $K'$ are extensions of the same field $F$, is every isomorphism $K \to K'$ a $F$-isomorphism?

Algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$. Alternative proof?

Prove that $[\mathbb{Q}(\sqrt[4]{2}, \sqrt{3}):\mathbb{Q}] = 8$

Composite of field extensions

Proving the lemma $\sqrt2\notin\mathbb{Q}(\sqrt3,\sqrt5)$

Proof that $[\overline{\mathbb Q}:\mathbb R\cap\overline{\mathbb Q}]=2$

Noether's Normalization Lemma states something that appears to contradict the Nullstellensatz. Where have I gone wrong?

Examples in finite extensions.

Galois Group of splitting field of $x^4 - x -1$ over $\mathbb{Q}$?

Proof verification. Question about finite extension, minimal polynomial.

A doubt in understanding a proof in Galois Theory

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