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sage: Qp=pAdicField(7)
sage: R.<x>=Qp[]
sage: K.<a>=Qp.extension(x^2-5)
sage: OK=K.integer_ring()
sage: OK
7-adic Unramified Extension Ring in a defined by x^2 - 5

Note that 15 is a square in $\mathbb{Q}_p$ , so $\mathbb{Q}_p(\sqrt{15})$ is just $\mathbb{Q}_p$ .

~~sage: Qp=pAdicField(7) sage: R.<x>=Qp[] sage: K.<a>=Qp.extension(x^2-5) K.=Qp.extension(x^2-5) sage: OK=K.integer_ring() sage: OK 7-adic Unramified Extension Ring in a defined by x^2 - 5~~
5

Note that 15 is a square in ~~$\mathbb{Q}_p$ ,~~ $\mathbb{Q}_7$ , so ~~$\mathbb{Q}_p(\sqrt{15})$~~ $\mathbb{Q}_7(\sqrt{15})$ is just ~~$\mathbb{Q}_p$ .~~ $\mathbb{Q}_7$ .

sage: Qp=pAdicField(7) sage: R.<x>=Qp[] sage: K.=Qp.extension(x^2-5) sage: OK=K.integer_ring() sage: OK 7-adic Unramified Extension Ring in a defined by x^2 - 5

Note that 15 is a square in $\mathbb{Q}_7$ , so $\mathbb{Q}_7(\sqrt{15})$ is just $\mathbb{Q}_7$ .

sage: Qp=pAdicField(7) sage: R.<x>=Qp[] sage: K.=Qp.extension(x^2-5) sage: OK=K.integer_ring() sage: OK 7-adic Unramified Extension Ring in a defined by x^2 - 5

Note that 15 is a square in $\mathbb{Q}_7$ , so $\mathbb{Q}_7(\sqrt{15})$ is just $\mathbb{Q}_7$ .

sage: Qp=pAdicField(7) sage: R.<x>=Qp[] sage: K.=Qp.extension(x^2-5) sage: OK=K.integer_ring() sage: OK 7-adic Unramified Extension Ring in a defined by x^2 - 5

Note that 15 is a square in $\mathbb{Q}_7$ , so $\mathbb{Q}_7(\sqrt{15})$ is just $\mathbb{Q}_7$ .

. sage: Qp=pAdicField(7) sage: R.<x>=Qp[] sage: K.=Qp.extension(x^2-5) sage: OK=K.integer_ring() sage: OK 7-adic Unramified Extension Ring in a defined by x^2 - 5

Note that 15 is a square in $\mathbb{Q}_7$ , so $\mathbb{Q}_7(\sqrt{15})$ is just $\mathbb{Q}_7$ .

. sage: Qp=pAdicField(7) sage: R.<x>=Qp[] sage: K.=Qp.extension(x^2-5) sage: OK=K.integer_ring() sage: OK 7-adic Unramified Extension Ring in a defined by x^2 - 5

Note that 15 is a square in $\mathbb{Q}_7$ , so $\mathbb{Q}_7(\sqrt{15})$ is just $\mathbb{Q}_7$ .

Try:

sage: Qp=pAdicField(7)
 sage: R.<x>=Qp[]
 sage: K.=Qp.extension(x^2-5)
    K.<a>=Qp.extension(x^2-5)
sage: OK=K.integer_ring()
 sage: OK
 7-adic Unramified Extension Ring in a defined by x^2 - 5
5

Note that 15 is a square in $\mathbb{Q}_7$ , so $\mathbb{Q}_7(\sqrt{15})$ is just $\mathbb{Q}_7$ .

Try:

sage: Qp=pAdicField(7)
sage: R.<x>=Qp[]
sage: K.<a>=Qp.extension(x^2-5)
sage: OK=K.integer_ring()
sage: OK
7-adic Unramified Extension Ring in a defined by x^2 - 5

Note that 15 is a square in $\mathbb{Q}_7$ , so ~~$\mathbb{Q}_7(\sqrt{15})$~~ $\mathbb{Q}_7(\sqrt{3}.\sqrt{5})$ is just ~~$\mathbb{Q}_7$ .~~ $\mathbb{Q}_7(\sqrt{5})$ .

Try:

sage: Qp=pAdicField(7)
sage: R.<x>=Qp[]
sage: K.<a>=Qp.extension(x^2-5)
sage: OK=K.integer_ring()
sage: OK
7-adic Unramified Extension Ring in a defined by x^2 - 5

Note that 15 is a square in $\mathbb{Q}_7$ , so ~~$\mathbb{Q}_7(\sqrt{3}.\sqrt{5})$~~ $\mathbb{Q}_7(\sqrt{3},\sqrt{5})$ is just $\mathbb{Q}_7(\sqrt{5})$ .

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