Nepal Algebra Project (NAP) / नेपाल बीज-गणित परियोजना


Module - IV : June 26, 2016 – July 08, 2016, Michel Waldschmidt
Topics: Computing Galois Groups, When is Gf contained in An? When is Gf transitive? Polynomials of degree at most three, Quartic poly-nomials, Examples of polynomials with Sp as Galois group over Q, Finite fields, Computing Galois groups over Q
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  • Correction of the mid term exam.
  • Examples of reducible polynomials without a root in the field.
  • A finite extension is algebraic, however there are algebraic extensions which are not finite.
  • A cyclic subgroup may or may not be normal.
  • If $f\in F[X]$ is a polynomial such that $f(\alpha)=0$, this does not imply that $f$ is the minimal polynomial of $\alpha$ over $F$.
  • $[\mathbf{Q}(e^{2i\pi/n}):$ $\mathbf{Q}]=n-1$ only when $n$ is prime.
  • In characteristic $p$, $X^{p^n}-X=$ $X(X^{p^n-1}-1)$ is not equal to $X(X-1)^{p^n-1}$.
  • The theorems of the course should be used, not proved again: an extension of degree $n$ of the prime field $\mathbf{F_p}$ has $p^n$ elements. The results on constructible numbers should also be used, not reproved.

Separable, normal, Galois extensions. Examples and counter examples.
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  • Examples of normal and of non normal extensions. Stem field, splitting field.
  • Quadratic extensions. Galois group when separable.
  • Galois group of a polynomial over a field as a subgroup of $\mathfrak {S_n}$.
    Example: the two polynomials $f_1=$ $X^4-8X^2+15$ and $f_2(X)=$ $X^4-16X^2+4$ have the same splitting field over $\mathbf{Q}$, since
    $ f_1=$ $(X^2-3)$ $(X^2-5)$
    $f_2=$ $(X-\sqrt{5}-\sqrt{3})$ $(X-\sqrt{5}+\sqrt{3})$ $(X+\sqrt{5}-\sqrt{3})$ $(X+\sqrt{5}+\sqrt{3});$
    however, as subgroups of $\mathfrak {S_4}$, the Galois group of $f_1$ over $\mathbf{Q}$ is transitive but not the Galois group of $f_2$.

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  • The symmetric group $\mathfrak{S_n}$ and the alternating group $\mathfrak{A_n}$.The symmetric group $\mathfrak{S_n}$. Examples: $n=1,2,3,4$. Order. Transpositions, cycles. Decomposition into disjoint cycles. Relation
    $(a_1,a_2)(a_2,a_3)\cdots$ $(a_{k-1},a_k)=$ $(a_1,a_2,\dots,a_k).$
  • Inversion, number of inversions, signature:sign = (−1)# inversions.
  • Group acting on a set: definition. Any group $G$ is isomorphic to a subgroup of the symmetric group $\mathfrak{S_G}$. In particular any finite group of order $n$ is isomorphic to a subgroup of $\mathfrak{S_n}$.
  • Action of $\mathfrak{S_n}$ on the set of maps $\mathbf{Z}^n\rightarrow \mathbf{Z}$
    $\sigma f(a_1,\dots,a_n)=$ $f(a_{\sigma(1)},\dots,a_{\sigma(n)}).$
    $ p(a_1,\dots,a_n)=$ $\prod_{1\le i\le n}(a_j-a_i).$

  • Consequence: $\sigma:\mathfrak{S_n}\rightarrow \{-1,+1\}$ is a group homomorphism, surjective for $n\ge 2$. Kernel: alternating group $\mathfrak{A_n}$.
  • Signature of a cycle of length $k$: $(-1)^{k-1}$.
  • Discriminant of a polynomial. Proposition 4.1 and Corollary 4.2 of Milne. Polynomials of degree $2$ and $3$: examples 4.5 and 4.6.

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  • Recall: Galois group of a separable polynomial of degree $\le 3$.
  • Study of the Galois group of a reducible polynomial of degree $4$. Examples: $(X^2+1)(X^2-2X+2)$ and $(X^2+1)(X^2-2)$.
  • Subgroups of $\mathfrak{S_4}$. Order and signature of a permutation decomposed into product of disjoint cycles.

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  • Galois groups over $\mathbf{Q}$ of
    • $\Phi_5=X^4$ $+X^3+X^2$ $+X+1:C_4$
    • $\Phi_8=X^4$ $+1:V_4$
    • $X^4-2:$ $D_4$
  • Transitive subgroups of $\mathfrak{S_4}$:
    $\mathfrak{S_4}, \; \mathfrak{A_4} , \; $ $ V_4, \; D_4,\; C_4.$
    Index of $V_4$ in each of them.
  • Resolvant.
  • Galois groups over $\mathbf{Q}$ of
    • $X^4+4X$ $+2:\mathfrak{S_4}$
    • $X^4+8X$ $+12:\mathfrak{A_4}$
    • References: Garling p.115–117. Milne p. 49–51.