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Let $R$ be the ring $$R = \prod_{p\ \text{prime}} \mathbb{F}_p$$ where $\mathbb{F}_p$ is the field having $p$ elements.

Is it true that $R$ has a quotient by a maximal ideal which is a field of characteristic zero and contains $\overline{\mathbb{Q}}$?

Motivation: I like the problem and I can't solve it... it should have something to do with the Chebotarev density theorem, but apparently other methods are possible, too! (See Joel's answer.)

Let $R$ be the ring $$R = \prod_{p\ \text{prime}} \mathbb{F}_p$$ where $\mathbb{F}_p$ is the field having $p$ elements.

Is it true that $R$ has a quotient by a maximal ideal which is a field of characteristic zero and contains $\overline{\mathbb{Q}}$?

Motivation: I like the problem and I can't solve it...

It should have something to do with the Chebotarev density theorem.

Let $R$ be the ring $$R = \prod_{p\ \text{prime}} \mathbb{F}_p$$ where $\mathbb{F}_p$ is the field having $p$ elements.

Is it true that $R$ has a quotient by a maximal ideal which is a field of characteristic zero and contains $\overline{\mathbb{Q}}$?

Motivation: I like the problem and I can't solve it...

Let $R$ be the ring $$R = \prod_{p\ \text{prime}} \mathbb{F}_p$$ where $\mathbb{F}_p$ is the field having $p$ elements.

Is it true that $R$ has a quotient by a maximal ideal which is a field of characteristic zero and contains $\overline{\mathbb{Q}}$?

Motivation: I like the problem and I can't solve it... it should have something to do with the Chebotarev density theorem, but apparently other methods are possible, too! (See Joel's answer.)