# Revision history [back]

### Finding a $\gamma$ to define a number field like $E(X^5-\gamma)$$E=\Q(\zeta_5)(X^5-\gamma) By working with eliptic curves, I found that the extension E defined by: E.< a> = NumberField(x^20 - 2x^19 - 2x^18 + 18x^17 - 32x^16 + 88x^15 + 58x^14 - 782x^13 + 1538x^12 + 1348x^11 - 466x^10 - 894x^9 + 346x^8 - 114x^7 - 424x^6 - 88x^5 + 214x^4 + 54x^3 + 14x^2 + 4*x + 1) Is a cyclic Kummer extension of degree 5 over \mathbb{Q}(\zeta_5), thus by the clasification of Kummer extensions, it exists \gamma \in \mathbb{Q}(\zeta_5) such that E=\mathbb{Q}(\sqrt{\gamma}). So, about all the polynomials f that define E how do I find \gamma such that the polynomial X^5-\gamma also define E? ### Finding a \gamma to define a number field like E=\Q(\zeta_5)(X^5-\gamma)$$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 $E=\mathbb{Q}(\sqrt{\gamma})$.

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 $E=\mathbb{Q}(\sqrt{\gamma})$.

So, about all the polynomials $f$ that define $E$ how do I find $\gamma$ such that the polynomial $X^5-\gamma$ also define $E$? 8 No.8 Revision

### 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 $E=\mathbb{Q}(\sqrt{\gamma})$.

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