"Let α be an algebraic integer and p(x) be its minimal (monic) polynomial p ( x ) = ∑ i = 0 n a i x i Such that p ( α ) = 0 and a i ∈ Z (Z is the set of integers) and a n = 1."The extension of a ring A by the element a is the set A [ α ] of all complex numbers of the form ∑ j = 0 n − 1 c j α j such that c j ∈ A, with all the operations inherited from A."The degree of the extension is the degree of the polynomial."I completely understand the 'algebraic integer' and the 'minimal polynomial' and the concept of set/ring extension, at least I think. My issue is mostly with the middle sentence; when it says 'all the complex numbers of form', but isn't there only one minimal polynomial so there's only one element in the set A [ α ]? Is that single element basically just p ( α ) − x n ? Or are the c j related to the a i at all? The same n is mentioned twice. If not then why introduce p(x) at all? And what is ring A? Is that an initially empty set? And what does 'all the operations inherited from A' mean? Honestly, I can't find an explanation online for the A [ α ] sort of notation or any of my other questions online?

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&quot;Let   α be an algebraic integer and p(x) be its minimal (monic) polynomial  p  (  x  )  =      ∑          i      =      0              n            a    i        x    i  Such that   p  (  α  )  =  0 and       a    i    ∈      Z   (Z is the set of integers) and       a    n    =  1.&quot;The extension of a ring A by the element a is the set   A  [  α  ] of all complex numbers of the form      ∑          j      =      0              n      −      1            c    j        α    j  such that       c    j    ∈  A, with all the operations inherited from A.&quot;The degree of the extension is the degree of the polynomial.&quot;I completely understand the &#039;algebraic integer&#039; and the &#039;minimal polynomial&#039; and the concept of set/ring extension, at least I think. My issue is mostly with the middle sentence; when it says &#039;all the complex numbers of form&#039;, but isn&#039;t there only one minimal polynomial so there&#039;s only one element in the set   A  [  α  ]? Is that single element basically just   p  (  α  )  −      x    n  ? Or are the       c    j   related to the       a    i   at all? The same n is mentioned twice. If not then why introduce p(x) at all? And what is ring A? Is that an initially empty set? And what does &#039;all the operations inherited from A&#039; mean? Honestly, I can&#039;t find an explanation online for the   A  [  α  ] sort of notation or any of my other questions online?

kuthiwenihca · Accepted Answer

Step 1Welcome to Mathematics Stack Exchange.A is a ring such as Z.  α is an algebraic integer such as i, whose minimal polynomial is       x    2    +  1.The extension   A  [  α  ] has numbers of the form       c    0    +      c    1    α  +      c    2        α    2    +  .  .  .  +      c          n      −      1            a          n      −      1        , with       c    j    ∈  A.In the example with   α  =  i  , the elements of Z[i] have the form       c    0    +      c    1    i.The       c    j   are not related to the       a    i  .Note that, if   α is a root of an       n          t      h       degree polynomial,then       α    n   can be expressed as a linear combination of   1  ,  α  ,      α    2    ,  .  .  .  ,      α          n      −      1      .In the example with   α  =  i,       α    2    =  −  1  (  1  )  +  0  (  α  ).That is why, in the sum for p(x), the index goes up to n,whereas in the sum for an element of   A  [  α  ]  , the index goes up to   n  −  1.Step 2Operations inherited from A means that when we add or multiply two elements of   A  [  α  ],say   (      c    0    +      c    1    α  +      c    2        α    2    +  .  .  .  +      c          n      −      1            α          n      −      1        )  +  (      d    0    +      d    1    α  +      d    2        α    2    +  .  .  .  +      d          n      −      1            α          n      −      1        )  ,the result is   (      c    0    +      d    0    )  +  (      c    1    +      d    1    )  α  +  (      c    2    +      d    2    )      α    2    +  .  .  .  +  (      c          n      −      1        +      d          n      −      1        )      α          n      −      1        ,where       c    j    +      d    j   is computed in A.And when we multiply   (      c    0    +      c    1    i  )  (      d    0    +      d    1    i  )  ,the result is       c    0        d    0    +  (      c    0        d    1    +      c    1        d    9    )  i  +      c    1        d    1        i    2    =      c    0        d    0    −      c    1        d    1    +  (      c    0        d    1    +      d    1        c    0    )  i  ,where again products and sums of terms involving       c    j   and       d    j   are computed in A.

Rhett Guerrero · Accepted Answer

Step 1I just want to expand on &quot;operations inherited from A.&quot; If the book only talks about infinite domains like Q and Z with the usual addition and multiplication, mentioning operation inheritance might be needlessly confusing.Consider for example the finite ring             Z              10      , consisting of only the integers 0 to 9. This would seem to have the usual addition and multiplication, since, for example,   1  +  1  =  1  ×  2  =  2.However,   7  +  7  =  7  ×  2 but does not equal 14. Both of those operations &quot;wrap around&quot; to land back in             Z              10      , giving 4 rather than 14 in this case.Step 2Now consider             Z              10        [      53    ]. Then       53    +      53    =  2      53   just like we expect. But   7  ×  2      53   is not   14      53   but   4      53  .What would   (      53        )    2   be? I&#039;m not exactly sure, I myself am confused on this point. But hopefully I have given you a clearer understanding of how addition and multiplication might differ from what you&#039;re used to.Rings of matrices might provide another good example.

"Let alpha be an algebraic integer and p(x) be its minimal (monic) polynomial p(x)=sum_{i=0}^{n}a_i x^i.

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