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This is a question related to this one. I'm working with sage and I'm trying to construct the following situation (perhaps some of you have done it beofre):

I'm trying to construct given an irreducible polynomial $f \in \mathbb{Q}[X]$ an extension field that adjoins two of its roots $\alpha_1,\alpha_2$. I'm trying to follow the approach suggested in this question . However, with the following code:

P.<x> = QQ[]
f = x^4+2*x+5 # f = P([5,2,0,1]) if you want
f_roots = f.roots(QQbar, multiplicities=False)
print f_roots
alpha = f_roots[0]
beta = f_roots[1]
K = QQ[alpha,beta]
K['x'](f).is_irreducible()

But this gives the error:

ValueError: defining polynomial (x^4 + 2*x + 5) must be irreducible

Although, the polynomial is clearly irreducible over $\mathbb{Q}$. Doing it as:

P.<x> = QQ[]
f = x^3+2*x+5
f_roots = f.roots(QQbar, multiplicities=False)
alpha = f_roots[0]
K.<a> = QQ[alpha]
beta = f_roots[1]
K1.<b> = K[beta]

Gives error:

ValueError: base field and extension cannot have the same name 'a'

What is going wrong? Is this the right way to construct the extension field with two roots?

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1 Answer 1

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One could define successive extensions, but the nicest might be to define individual extensions separately and to construct a composite field from those.

Define P, f, alpha, beta as in the question.

sage: P.<x> = QQ[]
sage: f = P([5, 2, 0, 0, 1])
sage: f_roots = f.roots(QQbar, multiplicities=False)
sage: f_roots
[-1.068757255260645? - 0.8212240798160035?*I,
 -1.068757255260645? + 0.8212240798160035?*I,
 1.068757255260645? - 1.268887367765514?*I,
 1.068757255260645? + 1.268887367765514?*I]
sage: alpha, beta = f_roots[:2]
sage: alpha, beta
(-1.068757255260645? - 0.8212240798160035?*I,
 -1.068757255260645? + 0.8212240798160035?*I)

Define the number fields $\mathbb{Q}[\alpha]$, $\mathbb{Q}[\beta]$ as embedded number fields, with embeddings in QQbar.

sage: K.<a> = NumberField(f, 'a', embedding=alpha)
sage: L.<b> = NumberField(f, 'b', embedding=beta)

Now the list of composite fields will contain the only compatible embedded number field. (Had K and L been constructed without embeddings, there would be several possible composite fields).

sage: K.composite_fields(L, 'c', both_maps=True)
[(Number Field in c with defining polynomial
      x^12 - 54*x^9 + 335*x^8 + 1900*x^6 - 17820*x^5 - 12725*x^4
      - 17928*x^3 + 421660*x^2 - 2103750*x + 6284221,
  Ring morphism:
    From: Number Field in a with defining polynomial x^4 + 2*x + 5
    To:   Number Field in c with defining polynomial
          x^12 - 54*x^9 + 335*x^8 + 1900*x^6 - 17820*x^5 - 12725*x^4
          - 17928*x^3 + 421660*x^2 - 2103750*x + 6284221
    Defn: a |--> -4787089492308500/19095694682980435222089*c^11
        - 60072370481993649/50921852487947827258904*c^10
        - 107347330062640525/25460926243973913629452*c^9
        + 54770246058614250/6365231560993478407363*c^8
        - 830196451346905669/19095694682980435222089*c^7
        - 6929297573110815045/25460926243973913629452*c^6
        - 80217555206153510375/38191389365960870444178*c^5
        + 32869508519243998767/25460926243973913629452*c^4
        + 218621879478829109530/19095694682980435222089*c^3
        + 3195870754505273678775/50921852487947827258904*c^2
        - 33579832334379607364857/76382778731921740888356*c
        + 315450855301503993255/1106996793216257114324,
  Ring morphism:
    From: Number Field in b with defining polynomial x^4 + 2*x + 5
    To:   Number Field in c with defining polynomial
          x^12 - 54*x^9 + 335*x^8 + 1900*x^6 - 17820*x^5 - 12725*x^4
          - 17928*x^3 + 421660*x^2 - 2103750*x + 6284221
    Defn: b |--> -9574178984617000/19095694682980435222089*c^11
        - 60072370481993649/25460926243973913629452*c^10
        - 107347330062640525/12730463121986956814726*c^9
        + 109540492117228500/6365231560993478407363*c^8
        - 1660392902693811338/19095694682980435222089*c^7
        - 6929297573110815045/12730463121986956814726*c^6
        - 80217555206153510375/19095694682980435222089*c^5
        + 32869508519243998767/12730463121986956814726*c^4
        + 437243758957658219060/19095694682980435222089*c^3
        + 3195870754505273678775/25460926243973913629452*c^2
        + 4611557031581263079321/38191389365960870444178*c
        + 315450855301503993255/553498396608128557162,
  -2)]

Extract this composite field.

sage: M.<c>, phi_KM, phi_LM, k = K.composite_fields(L, 'c', both_maps=True)[0]
sage: M
Number Field in c with defining polynomial
    x^12 - 54*x^9 + 335*x^8 + 1900*x^6 - 17820*x^5 - 12725*x^4
    - 17928*x^3 + 421660*x^2 - 2103750*x + 6284221
sage: M(a)
-4787089492308500/19095694682980435222089*c^11
- 60072370481993649/50921852487947827258904*c^10
- 107347330062640525/25460926243973913629452*c^9
+ 54770246058614250/6365231560993478407363*c^8
- 830196451346905669/19095694682980435222089*c^7
- 6929297573110815045/25460926243973913629452*c^6
- 80217555206153510375/38191389365960870444178*c^5
+ 32869508519243998767/25460926243973913629452*c^4
+ 218621879478829109530/19095694682980435222089*c^3
+ 3195870754505273678775/50921852487947827258904*c^2
- 33579832334379607364857/76382778731921740888356*c
+ 315450855301503993255/1106996793216257114324
sage: M(b)
-9574178984617000/19095694682980435222089*c^11
- 60072370481993649/25460926243973913629452*c^10
- 107347330062640525/12730463121986956814726*c^9
+ 109540492117228500/6365231560993478407363*c^8
- 1660392902693811338/19095694682980435222089*c^7
- 6929297573110815045/12730463121986956814726*c^6
- 80217555206153510375/19095694682980435222089*c^5
+ 32869508519243998767/12730463121986956814726*c^4
+ 437243758957658219060/19095694682980435222089*c^3
+ 3195870754505273678775/25460926243973913629452*c^2
+ 4611557031581263079321/38191389365960870444178*c
+ 315450855301503993255/553498396608128557162

Do some computations.

sage: QQbar(M(a)*M(b))
1.81665105994191? + 0.?e-14*I
sage: alpha*beta
1.81665105994191? + 0.?e-14*I
sage: QQbar(M(a)*M(b)) == alpha*beta
True
sage: QQbar(M(a) + M(b))
-2.137514510521290? + 0.?e-15*I
sage: alpha + beta
-2.137514510521290? + 0.?e-15*I
sage: QQbar(M(a) + M(b)) == alpha + beta
True

Further reading:

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