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Finding a $\gamma$ to define a number field like $E(X^5-\gamma)$

By working with eliptic curves, I found that the extension E defined by:

E.=NumberField(x^20 - 2x^19 - 2x^18 + 18x^17 - 32x^16 + 88x^15 + 58x^14 - 782x^13 + 1538x^12 + 1348x^11 - 466x^10 - 894x^9 + 346x^8 - 114x^7 - 424x^6 - 88x^5 + 214x^4 + 54x^3 + 14x^2 + 4*x + 1)

Is a cyclic Kummer extension of degree $5$ over $\mathbb{Q}(\zeta_5)$, thus by the clasification of Kummer extensions, it exists $\gamma \in \mathbb{Q}(\zeta_5)$ such that $E=\mathbb{Q}(\sqrt[5]{\zeta_5}$.

So, about all the polynomials $f$ that define $E$ how do I find $\gamma$ such that $the polynomial $X^5-\gamma$ also define $E$?

Finding a $\gamma$ to define a number field like $E(X^5-\gamma)$

By working with eliptic curves, I found that the extension E defined by:

E.=NumberField(x^20 - 2x^19 - 2x^18 + 18x^17 - 32x^16 + 88x^15 + 58x^14 - 782x^13 + 1538x^12 + 1348x^11 - 466x^10 - 894x^9 + 346x^8 - 114x^7 - 424x^6 - 88x^5 + 214x^4 + 54x^3 + 14x^2 + 4*x + 1)

Is a cyclic Kummer extension of degree $5$ over $\mathbb{Q}(\zeta_5)$, thus by the clasification of Kummer extensions, it exists $\gamma \in \mathbb{Q}(\zeta_5)$ such that ~~$E=\mathbb{Q}(\sqrt[5]{\zeta_5}$.~~$E=\mathbb{Q}(\sqrt[5]{\gamma})$.

So, about all the polynomials $f$ that define $E$ how do I find $\gamma$ such that $the polynomial $X^5-\gamma$ also define $E$?

Finding a $\gamma$ to define a number field like $E(X^5-\gamma)$

By working with eliptic curves, I found that the extension E defined by:

E.=NumberField(x^20 - 2x^19 - 2x^18 + 18x^17 - 32x^16 + 88x^15 + 58x^14 - 782x^13 + 1538x^12 + 1348x^11 - 466x^10 - 894x^9 + 346x^8 - 114x^7 - 424x^6 - 88x^5 + 214x^4 + 54x^3 + 14x^2 + 4*x + 1)

Is a cyclic Kummer extension of degree $5$ over $\mathbb{Q}(\zeta_5)$, thus by the clasification of Kummer extensions, it exists $\gamma \in \mathbb{Q}(\zeta_5)$ such that $E=\mathbb{Q}(\sqrt[5]{\gamma})$.

So, about all the polynomials $f$ that define $E$ how do I find $\gamma$ such that $the polynomial $X^5-\gamma$ also define $E$?

Finding a $\gamma$ to define a number field like $E(X^5-\gamma)$

By working with eliptic curves, I found that the extension E defined by:

~~E.=NumberField(x^20~~ E.< a> = NumberField(x^20 - 2x^19 - 2x^18 + 18x^17 - 32x^16 + 88x^15 + 58x^14 - 782x^13 + 1538x^12 + 1348x^11 - 466x^10 - 894x^9 + 346x^8 - 114x^7 - 424x^6 - 88x^5 + 214x^4 + 54x^3 + 14x^2 + 4*x + 1)

Is a cyclic Kummer extension of degree $5$ over $\mathbb{Q}(\zeta_5)$, thus by the clasification of Kummer extensions, it exists $\gamma \in \mathbb{Q}(\zeta_5)$ such that $E=\mathbb{Q}(\sqrt[5]{\gamma})$.

So, about all the polynomials $f$ that define $E$ how do I find $\gamma$ such that ~~$the~~ the polynomial $X^5-\gamma$ also define $E$?

Finding a $\gamma$ to define a number field like $E(X^5-\gamma)$$E=\Q(\zeta_5)(X^5-\gamma)$

By working with eliptic curves, I found that the extension E defined by:

E.< a> = NumberField(x^20 - 2x^19 - 2x^18 + 18x^17 - 32x^16 + 88x^15 + 58x^14 - 782x^13 + 1538x^12 + 1348x^11 - 466x^10 - 894x^9 + 346x^8 - 114x^7 - 424x^6 - 88x^5 + 214x^4 + 54x^3 + 14x^2 + 4*x + 1)

Is a cyclic Kummer extension of degree $5$ over $\mathbb{Q}(\zeta_5)$, thus by the clasification of Kummer extensions, it exists $\gamma \in \mathbb{Q}(\zeta_5)$ such that $E=\mathbb{Q}(\sqrt[5]{\gamma})$.

So, about all the polynomials $f$ that define $E$ how do I find $\gamma$ such that the polynomial $X^5-\gamma$ also define $E$?

Finding a $\gamma$ to define a number field like $E=\Q(\zeta_5)(X^5-\gamma)$$E=\mathbb{Q}(\zeta_5)(X^5-\gamma)$

By working with eliptic curves, I found that the extension E defined by:

E.< a> = NumberField(x^20 - 2x^19 - 2x^18 + 18x^17 - 32x^16 + 88x^15 + 58x^14 - 782x^13 + 1538x^12 + 1348x^11 - 466x^10 - 894x^9 + 346x^8 - 114x^7 - 424x^6 - 88x^5 + 214x^4 + 54x^3 + 14x^2 + 4*x + 1)

Is a cyclic Kummer extension of degree $5$ over $\mathbb{Q}(\zeta_5)$, thus by the clasification of Kummer extensions, it exists $\gamma \in \mathbb{Q}(\zeta_5)$ such that $E=\mathbb{Q}(\sqrt[5]{\gamma})$.

So, about all the polynomials $f$ that define $E$ how do I find $\gamma$ such that the polynomial $X^5-\gamma$ also define $E$?

Finding a $\gamma$ to define a number field like $E=\mathbb{Q}(\zeta_5)(X^5-\gamma)$

By working with eliptic curves, I found that the extension E defined by:

E.< a> = NumberField(x^20 - 2x^19 - 2x^18 + 18x^17 - 32x^16 + 88x^15 + 58x^14 - 782x^13 + 1538x^12 + 1348x^11 - 466x^10 - 894x^9 + 346x^8 - 114x^7 - 424x^6 - 88x^5 + 214x^4 + 54x^3 + 14x^2 + 4*x + 1)

Is a cyclic Kummer extension of degree $5$ over $\mathbb{Q}(\zeta_5)$, thus by the clasification of Kummer extensions, it exists $\gamma \in \mathbb{Q}(\zeta_5)$ such that $E=\mathbb{Q}(\sqrt[5]{\gamma})$.

So, about all the polynomials $f$ that define $E$ how do I find $\gamma$ such that the polynomial $X^5-\gamma$ also define $E$?

Finding a $\gamma$ to define a number field like $E=\mathbb{Q}(\zeta_5)(X^5-\gamma)$

By working with eliptic curves, I found that the extension E defined by:

E.< a>
E.<a> = NumberField(x^20 - 2x^19 2*x^19 - 2*x^18 + 18*x^17 - 32*x^16 + 88*x^15 + 58*x^14 - 2x^18 + 18x^17 - 32x^16 + 88x^15 + 58x^14 - 782x^13 + 1538x^12 + 1348x^11 - 466x^10 - 894x^9 + 346x^8 - 114x^7 - 424x^6 - 88x^5 + 214x^4 782*x^13 + 54x^3 1538*x^12 + 14x^2 1348*x^11 - 466*x^10 - 894*x^9 + 346*x^8 - 114*x^7 - 424*x^6 - 88*x^5 + 214*x^4 + 54*x^3 + 14*x^2 + 4*x + 1)

1)

Is a cyclic Kummer extension of degree $5$ over $\mathbb{Q}(\zeta_5)$, thus by the clasification of Kummer extensions, it there exists $\gamma \in \mathbb{Q}(\zeta_5)$ such that $E=\mathbb{Q}(\sqrt[5]{\gamma})$.

So, about all the polynomials $f$ that define $E$ how do I find $\gamma$ such that the polynomial $X^5-\gamma$ also ~~define~~ defines $E$?

Revision history [back]

Finding a $\gamma$ to define a number field like $E(X^5-\gamma)$

Finding a $\gamma$ to define a number field like $E(X^5-\gamma)$

Finding a $\gamma$ to define a number field like $E(X^5-\gamma)$

Finding a $\gamma$ to define a number field like $E(X^5-\gamma)$

Finding a $\gamma$ to define a number field like $E(X^5-\gamma)$$E=\Q(\zeta_5)(X^5-\gamma)$

Finding a $\gamma$ to define a number field like $E=\Q(\zeta_5)(X^5-\gamma)$$E=\mathbb{Q}(\zeta_5)(X^5-\gamma)$

Finding a $\gamma$ to define a number field like $E=\mathbb{Q}(\zeta_5)(X^5-\gamma)$

Finding a $\gamma$ to define a number field like $E=\mathbb{Q}(\zeta_5)(X^5-\gamma)$