# Sage thinks that I^(2/3) = -1

Reported by: Owned by: mmezzarobba major sage-9.3 basic arithmetic gh-BrentBaccala Marc Mezzarobba Sébastien Labbé N/A 0807787 080778741065560bdce8be7a4c207e97ab0163d4

Revert #25218.

Rationale: #25218 makes `a^(p/q)` where a is a number field element attempt to return a solution to `a^p = x^q` in the number field. This makes powering inconsistent with coercions, since number fields often (and, in the case of quadratic fields, automatically) come with a complex embedding and complex numbers of various kinds use the principal branch of the power function.

Among other confusing behaviors, we have

```sage: I^(2/3)
I^(2/3)
sage: QQbar(I^(2/3))
0.500000000000000? + 0.866025403784439?*I
sage: QQbar(I)^(2/3)
0.500000000000000? + 0.866025403784439?*I
```

but

```sage: I.pyobject()^(2/3)
-1
```

and

```sage: QQi.<i> = QuadraticField(-1)
sage: i^(2/3)
-1
```

but

```sage: QQbar(i)^(2/3)
0.500000000000000? + 0.866025403784439?*I
```

as well as

```sage: CC((i^(1/6))^2)
0.866025403784439 + 0.500000000000000*I
sage: CC((i^(1/3)))
-1.00000000000000*I
sage: i^(1/3)*CC(i)^(1/3)
0.500000000000000 - 0.866025403784439*I
```

### comment:1 Changed 21 months ago by tmonteil

```sage: I^(2/3)
I^(2/3)
sage: _.n()
0.500000000000000 + 0.866025403784439*I
```

### comment:2 Changed 21 months ago by mmezzarobba

Sorry, I didn't have time yesterday. This happens for i ∈ ℚ[i], not i ∈ SR (see the linked ticket). But note that #18036 aims to make the glbal symbol i refer to the generator of ℚ[i]. (In fact, I am currently trying again to make that change, and that's how I discovered the issue.)

### comment:3 Changed 21 months ago by slabbe

Oups, I made a mistake while reviewing #25218. Feel free to fix that to the correct thing.

### comment:4 Changed 21 months ago by mmezzarobba

• Authors set to Marc Mezzarobba
• Branch set to u/mmezzarobba/30783-revert_25218
• Commit set to 080778741065560bdce8be7a4c207e97ab0163d4
• Description modified (diff)
• Status changed from new to needs_review

New commits:

 ​38771d3 `Revert "25218: space fixes"` ​3aae21d `Revert "Trac #25218: fix typo"` ​0807787 `Revert "Trac #25218: NumberField attempts to evaluate fractional powers"`

### comment:5 Changed 21 months ago by slabbe

• Status changed from needs_review to positive_review

I confirm with the branch I get this:

```sage: QQi.<i> = QuadraticField(-1)
sage: i^(2/3)
I^(2/3)
sage: n(_)
0.500000000000000 + 0.866025403784439*I
```

Bot is green (except for startup time which I think we can ignore because this ticket does not change anything regarding to that).

### comment:6 Changed 21 months ago by slabbe

• Reviewers set to Sébastien Labbé

### comment:7 Changed 21 months ago by mkoeppe

• Milestone changed from sage-9.2 to sage-9.3

### comment:8 Changed 20 months ago by vbraun

• Branch changed from u/mmezzarobba/30783-revert_25218 to 080778741065560bdce8be7a4c207e97ab0163d4
• Resolution set to fixed
• Status changed from positive_review to closed
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